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Math Formulae Page: 440 K-12 Formulas

A complete K-12 math formulae reference with 440 formulas across arithmetic, algebra, geometry, trigonometry, statistics, finance math, matrices, vectors, sets, and AP or IB calculus readiness.

Published: May 17, 2026Updated: May 17, 2026

Guide Oversight & Review Policy

CalculatorWallah guides are written to explain calculator assumptions, source limitations, and when users should move from a rough estimate to an official rule, institution policy, or clinician conversation.

Reviewed by Jitendra Kumar, Founder & Editorial Standards Lead. Page updated May 17, 2026. Trust-critical pages are reviewed when official rates or rules change. Evergreen calculator guides are checked on a recurring quarterly or annual cycle depending on topic volatility. Topic ownership: Sales tax and tax-sensitive estimate tools, Education and GPA planning calculators, Health, protein, and screening-formula pages, Platform-wide publishing standards and methodology.

Sources & methodology · Review standards

Overview

This math formulae page turns the uploaded K-12 checklist into a complete reference: 440 formula entries across 26 topics. It starts with number sense, fractions, ratios, exponents, and algebra, then moves into coordinate geometry, 2D and 3D geometry, trigonometry, statistics, financial mathematics, matrices, vectors, logic, and calculus readiness.

The goal is not to make students memorize a wall of symbols. A formula page is useful when it helps you identify the right relationship, check units, substitute values carefully, and decide whether a calculator or graphing step should come next. For formula-heavy solving, pair this reference with the Math Equation Solver, Scientific Calculator, and topic-specific tools in the Math Calculators hub.

How To Use This Page

Start with the topic, not the formula name. If the problem is about a triangle, go to the triangle section before searching for an equation. If it is about a sequence, decide whether the pattern is arithmetic, geometric, recursive, finite, or infinite before substituting numbers.

Then read the "Use" column. Similar-looking formulas often answer different questions: circumference is distance around a circle, circle area is space inside the circle, arc length is part of the circumference, and sector area is part of the circle's area. In algebra, slope, average rate of change, secant slope, and difference quotient all look related, but they sit at different levels of precision.

Finally, check restrictions. Denominators cannot be zero, logarithm inputs must be positive, inverse functions require one-to-one behavior, geometric infinite series need |r| < 1, and many geometry formulas assume matching units. The table keeps those conditions short so the formula remains scannable. Each row also includes a variables/notation column so the symbols are defined beside the MathJax-rendered formula.

Educational Video

I looked for a credible supporting video from an educational or institutional source rather than embedding a random creator video. A suitable Khan Academy video exists for the foundation behind many geometry formulas, so it is embedded here as context before the complete formula list.

This Khan Academy video is relevant because it shows how formulas start from meaning: perimeter as distance around a shape and area as covered space. That context matters before students jump into the full formula table below.

Formula Index

Use this index to jump to the right section. The full table below keeps the uploaded numbering from 1 through 440, so every item in the checklist has a matching formula entry.

1. Number Sense & Arithmetic30 formulae 2. Factors, Multiples & Number Theory19 formulae 3. Integers & Rational Numbers12 formulae 4. Algebra Basics13 formulae 5. Linear Equations & Functions18 formulae 6. Systems of Equations9 formulae 7. Exponents, Radicals & Roots11 formulae 8. Polynomials17 formulae 9. Quadratic Equations & Functions14 formulae 10. Functions16 formulae 11. Exponential & Logarithmic Functions15 formulae 12. Sequences & Series11 formulae 13. Coordinate Geometry15 formulae 14. Basic Geometry16 formulae 15. Triangles16 formulae 16. Quadrilaterals16 formulae 17. Circles14 formulae 18. Three-Dimensional Geometry19 formulae 19. Measurement12 formulae 20. Trigonometry20 formulae 21. Data, Statistics & Probability33 formulae 22. Financial Mathematics18 formulae 23. Calculus Readiness / Precalculus16 formulae 24. Matrices & Vectors18 formulae 25. Logic, Sets & Discrete Math15 formulae 26. Advanced K-12 / AP & IB Extension27 formulae

1. Number Sense & Arithmetic (30 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
1	Place value formula	\[N=\sum_{i=0}^{k}d_i\times10^i\]	\(N\): total count, population size, or original number \(d\): distance, diameter, divisor, or common difference \(i\): index value or matrix row index	Break a number into digit values by powers of 10.
2	Expanded form formula	\[N=a_n10^n+\cdots+a_1\cdot10+a_0\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(N\): total count, population size, or original number \(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Write a number as the sum of its place-value parts.
3	Standard form formula	\[N=\sum \text{place values}\]	\(N\): total count, population size, or original number	Combine expanded place values into one ordinary numeral.
4	Rounding rule	\[\begin{gathered}\text{next digit >= 5 -> round up}\\\text{next digit < 5 -> keep}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Round a value to a selected place.
5	Absolute value formula	\[\|x\|=\begin{cases}x,&x\ge0\\-x,&x<0\end{cases}\]	\(x\): unknown, input variable, or x-coordinate	Find distance from zero without sign direction.
6	Additive inverse formula	\[a + (-a) = 0\]	\(a\): a known number, coefficient, side length, or constant	Find the number that cancels a value under addition.
7	Multiplicative inverse formula	\[a\times\frac{1}{a}=1,\quad a\ne0\]	\(a\): a known number, coefficient, side length, or constant	Find the reciprocal that cancels a value under multiplication.
8	Order of operations rule	\[\begin{gathered}\text{Parentheses -> exponents -> multiply/divide -> add/subtract}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Evaluate expressions in the standard order.
9	Fraction addition formula	\[\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Add fractions with unlike denominators.
10	Fraction subtraction formula	\[\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Subtract fractions with unlike denominators.
11	Fraction multiplication formula	\[\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Multiply numerators and denominators.
12	Fraction division formula	\[\frac{a/b}{c/d}=\frac{ad}{bc}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Divide by multiplying by the reciprocal.
13	Mixed number to improper fraction formula	\[a\frac{b}{c}=\frac{ac+b}{c}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant	Convert a whole-number-plus-fraction into one fraction.
14	Improper fraction to mixed number formula	\[\frac{a}{b}=q+\frac{r}{b},\quad a=bq+r\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(q\): quotient, probability of failure, or integer part \(r\): radius, rate, ratio, remainder, or root	Separate an improper fraction into whole part and remainder.
15	Equivalent fractions formula	\[\frac{a}{b}=\frac{ak}{bk},\quad k\ne0\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(k\): constant, scale factor, index, or number of successes	Create equal fractions by scaling numerator and denominator.
16	Simplifying fractions formula	\[\frac{a}{b}=\frac{a/g}{b/g},\quad g=\gcd(a,b)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(g\): function g, greatest common factor, or gravitational/constant term	Reduce a fraction to lowest terms.
17	Decimal to fraction conversion formula	\[\text{decimal}=\frac{\text{integer}}{10^n}\]	\(n\): number of terms, sides, observations, trials, or periods	Convert a terminating decimal with n decimal places to a fraction.
18	Fraction to decimal conversion formula	\[a/b = a ÷ b\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Convert a fraction by dividing numerator by denominator.
19	Percent to decimal conversion formula	\[p\%=\frac{p}{100}\]	\(p\): rate, probability, or focus distance	Turn a percent into a decimal multiplier.
20	Decimal to percent conversion formula	\[\text{percent}=100\times\text{decimal}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Turn a decimal into a percent.
21	Fraction to percent conversion formula	\[\frac{a}{b}\times100\%\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Convert a fraction into a percent.
22	Percent of a number formula	\[\text{part}=\frac{p}{100}\times\text{whole}\]	\(p\): rate, probability, or focus distance	Find p percent of a quantity.
23	Percent increase formula	\[\text{new}=\text{original}\times\left(1+\frac{p}{100}\right)\]	\(p\): rate, probability, or focus distance	Increase a value by p percent.
24	Percent decrease formula	\[\text{new}=\text{original}\times\left(1-\frac{p}{100}\right)\]	\(p\): rate, probability, or focus distance	Decrease a value by p percent.
25	Percent change formula	\[\text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times100\%\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure relative change from an original value.
26	Ratio formula	\[a:b = a / b\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Compare two quantities by division.
27	Proportion formula	\[\frac{a}{b}=\frac{c}{d}\Longrightarrow ad=bc\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Solve equal-ratio problems by cross multiplication.
28	Unit rate formula	\[\text{unit rate}=\frac{\text{quantity}}{1\ \text{unit}}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Express a rate per one unit.
29	Average rate formula	\[\text{average rate}=\frac{\text{total change}}{\text{total time}}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find one constant rate over an interval.
30	Scale factor formula	\[\text{scale factor}=\frac{\text{new length}}{\text{original length}}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Compare similar figures or resized quantities.

2. Factors, Multiples & Number Theory (19 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
31	Divisibility rules	\[\begin{gathered}\text{n is divisible by d if n mod d = 0}\end{gathered}\]	\(d\): distance, diameter, divisor, or common difference \(n\): number of terms, sides, observations, trials, or periods Rule: Read the text condition before applying the symbols or calculator workflow.	Test whether division leaves no remainder.
32	Factor formula	\[n=a\times b\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Identify numbers that multiply to make n.
33	Multiple formula	\[m=k\times n\]	\(k\): constant, scale factor, index, or number of successes \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Generate multiples of a number.
34	Prime factorization formula	\[n = p_1^{a} p_2^{b} ...\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods \(p\): rate, probability, or focus distance	Write a number as powers of primes.
35	Greatest common factor formula	\[\begin{gathered}\text{gcf = product of common primes with smaller exponents}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find the largest shared factor.
36	Least common multiple formula	\[\operatorname{lcm}(a,b)=\frac{\|ab\|}{\gcd(a,b)}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Find the smallest shared multiple.
37	Euclidean algorithm formula	\[\gcd(a,b)=\gcd(b,a\bmod b)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Find a GCF by repeated remainders.
38	Exponent rules	\[a^m a^n=a^{m+n},\quad (a^m)^n=a^{mn}\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Simplify powers with the same base.
39	Product of powers rule	\[a^m\times a^n=a^{m+n}\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Multiply powers with the same base.
40	Quotient of powers rule	\[\frac{a^m}{a^n}=a^{m-n},\quad a\ne0\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Divide powers with the same base.
41	Power of a power rule	\[(a^m)^n=a^{mn}\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Raise one power to another power.
42	Power of a product rule	\[(ab)^n=a^n b^n\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Distribute an exponent across multiplication.
43	Power of a quotient rule	\[\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n},\quad b\ne0\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Distribute an exponent across division.
44	Zero exponent rule	\[a^0=1,\quad a\ne0\]	\(a\): a known number, coefficient, side length, or constant	Simplify any nonzero base raised to zero.
45	Negative exponent rule	\[a^{-n}=\frac{1}{a^n},\quad a\ne0\]	\(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Rewrite negative exponents as reciprocals.
46	Scientific notation formula	\[N=a\times10^n,\quad 1\le \|a\|<10\]	\(N\): total count, population size, or original number \(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Write very large or small numbers compactly.
47	Standard form to scientific notation formula	\[\text{move decimal }k\text{ places}\to a\times10^k\]	\(a\): a known number, coefficient, side length, or constant \(k\): constant, scale factor, index, or number of successes	Convert an ordinary number to scientific notation.
48	Scientific notation multiplication rule	\[(a\times10^m)(b\times10^n)=ab\times10^{m+n}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Multiply numbers written in scientific notation.
49	Scientific notation division rule	\[\frac{a\times10^m}{b\times10^n}=\frac{a}{b}\times10^{m-n}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Divide numbers written in scientific notation.

3. Integers & Rational Numbers (12 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
50	Integer addition rule	\[\begin{gathered}\text{same signs: add absolute values}\\\text{different signs: subtract absolute values}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Add signed whole numbers.
51	Integer subtraction rule	\[a - b = a + (-b)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Turn subtraction into adding the opposite.
52	Integer multiplication rule	\[\begin{gathered}\text{(+)(+) = +, (-)(-) = +, (+)(-) = -}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Choose the sign when multiplying integers.
53	Integer division rule	\[\begin{gathered}\text{same signs -> positive}\\\text{different signs -> negative}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Choose the sign when dividing integers.
54	Rational number addition formula	\[\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Add rational numbers as fractions.
55	Rational number subtraction formula	\[\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Subtract rational numbers as fractions.
56	Rational number multiplication formula	\[\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Multiply rational numbers.
57	Rational number division formula	\[\frac{a/b}{c/d}=\frac{ad}{bc},\quad c\ne0\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Divide rational numbers.
58	Comparing rational numbers formula	\[a/b ? c/d by comparing ad ? bc\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Compare fractions with positive denominators.
59	Ordering rational numbers rule	\[\begin{gathered}\text{use common denominators or decimal values}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Place rational numbers from least to greatest.
60	Opposite numbers formula	\[a + (-a) = 0\]	\(a\): a known number, coefficient, side length, or constant	Pair a number with its additive inverse.
61	Reciprocal formula	\[reciprocal of a = 1/a, a \ne 0\]	\(a\): a known number, coefficient, side length, or constant	Find the multiplicative inverse.

4. Algebra Basics (13 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
62	Variable expression formula	\[\begin{gathered}\text{expression value = expression after substituting variables}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Represent unknown quantities with letters.
63	Algebraic expression evaluation formula	\[\begin{gathered}\text{if x = a, replace x with a and simplify}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant	Evaluate an expression for given values.
64	Combining like terms rule	\[\begin{gathered}\text{ax + bx = (a + b)x}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Combine terms with the same variable part.
65	Distributive property formula	\[a(b + c) = ab + ac\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant	Multiply across a sum or difference.
66	Factoring common factor formula	\[ab + ac = a(b + c)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant	Pull out a shared factor.
67	One-step equation formula	\[x+a=b\Longrightarrow x=b-a,\quad ax=b\Longrightarrow x=\frac{b}{a}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Undo one operation to solve.
68	Two-step equation formula	\[ax+b=c\Longrightarrow x=\frac{c-b}{a}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant	Undo addition/subtraction, then multiplication/division.
69	Multi-step equation formula	\[\begin{gathered}\text{simplify both sides, then isolate x}\end{gathered}\]	\(x\): unknown, input variable, or x-coordinate	Solve equations with combining, distributing, or repeated operations.
70	Literal equation rearrangement formula	\[\begin{gathered}\text{solve for target variable using inverse operations}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Rearrange formulas for a different variable.
71	Inequality solving rule	\[\begin{gathered}\text{multiply/divide by a negative -> reverse inequality sign}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant	Solve inequalities without losing order direction.
72	Compound inequality formula	\[\begin{gathered}\text{a < x < b or x < a OR x > b}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Solve "and" and "or" inequality statements.
73	Absolute value equation formula	\[\|x-a\|=b\Longrightarrow x=a\pm b,\quad b\ge0\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Solve equations involving distance from a center.
74	Absolute value inequality formula	\[\begin{gathered}\|x-a\|<b\Longrightarrow a-b<x<a+b\\\|x-a\|>b\Longrightarrow x<a-b\ \text{or}\ x>a+b\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Solve distance-bounded inequalities.

5. Linear Equations & Functions (18 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
75	Slope formula	\[m=\frac{y_2-y_1}{x_2-x_1}\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(m\): slope \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find steepness from two points.
76	Slope-intercept form	\[y=mx+b\]	\(b\): b known number, coefficient, side length, or constant \(y\): dependent variable, output, or y-coordinate	Write a line using slope and y-intercept.
77	Point-slope form	\[y-y_1=m(x-x_1)\]	\(x_1\): x-1 coordinate of a point \(y_1\): y-1 coordinate of a point \(m\): slope \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Write a line from one point and slope.
78	Standard form of a line	\[Ax + By = C\]	\(C\): constant, coefficient, circumference, or third event as used in the row	Write a line with x and y terms on one side.
79	Two-point form of a line	\[y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Write a line through two known points.
80	Horizontal line formula	\[y = c\]	\(c\): c known number, coefficient, side length, or constant \(y\): dependent variable, output, or y-coordinate	Represent a line with zero slope.
81	Vertical line formula	\[x = c\]	\(c\): c known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Represent a line with undefined slope.
82	Parallel lines slope rule	\[m1 = m2\]	\(m\): slope	Check whether nonvertical lines are parallel.
83	Perpendicular lines slope rule	\[m_1m_2=-1\]	\(m\): slope	Check whether nonvertical lines meet at right angles.
84	x-intercept formula	\[\begin{gathered}\text{set y = 0 and solve for x}\end{gathered}\]	\(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find where a graph crosses the x-axis.
85	y-intercept formula	\[\begin{gathered}\text{set x = 0 and solve for y}\end{gathered}\]	\(y\): dependent variable, output, or y-coordinate	Find where a graph crosses the y-axis.
86	Linear function formula	\[f(x) = mx + b\]	\(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Model constant rate of change.
87	Direct variation formula	\[y = kx\]	\(y\): dependent variable, output, or y-coordinate	Model proportional relationships.
88	Inverse variation formula	\[y = k/x\]	\(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Model relationships where product xy is constant.
89	Constant rate of change formula	\[\begin{gathered}\text{rate = Delta y / Delta x}\end{gathered}\]	\(\Delta x\): width of each subinterval or change in x \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Measure equal change per input unit.
90	Arithmetic sequence formula	\[a_n=a_1+(n-1)d\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(a\): a known number, coefficient, side length, or constant \(d\): common difference \(n\): number of terms, sides, observations, trials, or periods	Find the nth term of a linear sequence.
91	Arithmetic series formula	\[S_n=\frac{n}{2}(a_1+a_n)\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(S\): sum, series total, sale price, or named set S \(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Sum the first n terms of an arithmetic sequence.
92	Linear interpolation formula	\[y=y_1+\frac{(x-x_1)(y_2-y_1)}{x_2-x_1}\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Estimate between two known points on a line.

6. Systems of Equations (9 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
93	System of linear equations formula	\[a1x + b1y = c1; a2x + b2y = c2\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Represent two linear equations solved together.
94	Substitution method formula	\[\begin{gathered}\text{solve one equation for a variable, then substitute into the other}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant	Solve a system by replacing a variable expression.
95	Elimination method formula	\[\begin{gathered}\text{multiply equations so one variable cancels when added/subtracted}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Solve a system by removing one variable.
96	Graphing method rule	\[\begin{gathered}\text{solution = intersection point(s)}\end{gathered}\]	\(s\): semiperimeter, arc length, side length, or sample standard deviation Rule: Read the text condition before applying the symbols or calculator workflow.	Solve a system by graphing.
97	Determinant formula for 2x2 systems	\[D = ad - bc\]	\(D\): determinant	Compute the determinant of [[a,b],[c,d]].
98	Cramer's rule for 2x2 systems	\[x=\frac{D_x}{D},\quad y=\frac{D_y}{D}\]	\(D_x\): determinant after replacing the matching variable column \(D_y\): determinant after replacing the matching variable column \(D\): determinant \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Solve a 2-variable system using determinants.
99	Cramer's rule for 3x3 systems	\[x_i = \det(A_i) / \det(A)\]	\(x_i\): ith data value or sample value \(A\): area, amount, angle, event, or matrix named A as used in the row \(i\): index value or matrix row index \(x\): unknown, input variable, or x-coordinate	Solve a 3-variable system using determinant replacement.
100	Matrix equation formula	\[Ax = b\]	\(b\): b known number, coefficient, side length, or constant	Write a linear system in matrix form.
101	Matrix inverse solution formula	\[x = A^(-1)b\]	\(A\): matrix or first event, depending on the row \(b\): b known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Solve an invertible matrix equation.

7. Exponents, Radicals & Roots (11 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
102	Square root formula	\[\sqrt{x}=y\Longleftrightarrow y^2=x,\ y\ge0\]	\(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find the principal square root.
103	Cube root formula	\[\sqrt[3]{x}=y\Longleftrightarrow y^3=x\]	\(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find the number whose cube is x.
104	nth root formula	\[\sqrt[n]{x}=x^{1/n}\]	\(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Rewrite roots as fractional exponents.
105	Radical simplification formula	\[\sqrt{ab} = \sqrt{a}\sqrt{b}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Separate perfect-square factors.
106	Radical multiplication rule	\[\sqrt{a}\sqrt{b} = \sqrt{ab}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Multiply compatible radicals.
107	Radical division rule	\[\sqrt{a}/\sqrt{b} = \sqrt{a/b}, b > 0\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Divide compatible radicals.
108	Rational exponent formula	\[a^(m/n) = \sqrt[n]{a^{m}}\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Connect exponents and roots.
109	Converting radicals to rational exponents	\[\sqrt[n]{a^{m}} = a^(m/n)\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Rewrite radical notation as exponent notation.
110	Converting rational exponents to radicals	\[a^(m/n) = \sqrt[n]{a^{m}}\]	\(a\): a known number, coefficient, side length, or constant \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Rewrite exponent notation as radical notation.
111	Square root property	\[x^2=a\Longrightarrow x=\pm\sqrt{a}\]	\(a\): a known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Solve quadratic equations that are perfect squares.
112	Distance using square roots formula	\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(d\): distance, diameter, divisor, or common difference \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Use the Pythagorean theorem on coordinate differences.

8. Polynomials (17 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
113	Polynomial degree rule	\[\begin{gathered}\text{degree = highest exponent with nonzero coefficient}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Identify polynomial degree.
114	Polynomial addition rule	\[(\sum a_i x^{i}) + (\sum b_i x^{i}) = \sum(a_i + b_i)x^{i}\]	\(a_i\): ith term or indexed value \(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(i\): index value or matrix row index \(x\): unknown, input variable, or x-coordinate	Add like polynomial terms.
115	Polynomial subtraction rule	\[P(x) - Q(x) = P(x) + (-Q(x))\]	\(P\): principal, perimeter, probability, or point P as used in the row \(Q\): quartile, matrix quotient, or named quantity Q \(x\): unknown, input variable, or x-coordinate	Subtract by distributing the negative.
116	Polynomial multiplication rule	\[(\sum a_i x^{i})(\sum b_j x^{j}) = \sum a_i b_j x^(i+j)\]	\(a_i\): ith term or indexed value \(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(i\): index value or matrix row index \(j\): index value or matrix column index \(x\): unknown, input variable, or x-coordinate	Multiply every term by every term.
117	Monomial multiplication formula	\[(ax^{m})(bx^{n}) = ab x^(m+n)\]	\(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Multiply one-term expressions.
118	Binomial multiplication formula	\[(a + b)(c + d) = ac + ad + bc + bd\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(c\): c coefficient in the polynomial or quadratic \(d\): distance, diameter, divisor, or common difference	Multiply two binomials.
119	FOIL formula	\[(a + b)(c + d) = ac + ad + bc + bd\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(c\): c coefficient in the polynomial or quadratic \(d\): distance, diameter, divisor, or common difference	Use first, outer, inner, last for binomials.
120	Special product: square of a binomial	\[(a\pm b)^2=a^2\pm2ab+b^2\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic	Expand a binomial square.
121	Special product: difference of squares	\[(a-b)(a+b)=a^2-b^2\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic	Factor or expand conjugates.
122	Special product: sum of cubes	\[a^3+b^3=(a+b)(a^2-ab+b^2)\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic	Factor a sum of cubes.
123	Special product: difference of cubes	\[a^3-b^3=(a-b)(a^2+ab+b^2)\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic	Factor a difference of cubes.
124	Factoring trinomials formula	\[x^{2} + bx + c = (x + m)(x + n), m + n = b, mn = c\]	\(b\): b coefficient in the polynomial or quadratic \(c\): c coefficient in the polynomial or quadratic \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Factor simple quadratic trinomials.
125	Factoring by grouping formula	\[ax + ay + bx + by = (a + b)(x + y)\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Factor four-term expressions by pairs.
126	Remainder theorem	\[\begin{gathered}\text{remainder when P(x) is divided by x - a is P(a)}\end{gathered}\]	\(P\): principal, perimeter, probability, or point P as used in the row \(a\): a coefficient in the polynomial or quadratic \(x\): unknown, input variable, or x-coordinate	Find a division remainder without dividing.
127	Factor theorem	\[x - a is a factor of P(x) \Longleftrightarrow P(a) = 0\]	\(P\): principal, perimeter, probability, or point P as used in the row \(a\): a coefficient in the polynomial or quadratic \(x\): unknown, input variable, or x-coordinate	Connect zeros and linear factors.
128	Polynomial long division formula	\[P(x) = D(x)Q(x) + R(x)\]	\(D\): difference, determinant, diameter, or named constant D \(P\): principal, perimeter, probability, or point P as used in the row \(Q\): quartile, matrix quotient, or named quantity Q \(R\): radius, right sum, range endpoint, or outer radius \(x\): unknown, input variable, or x-coordinate	Divide polynomials with quotient and remainder.
129	Synthetic division formula	\[\begin{gathered}\text{use coefficients with root a for divisor x - a}\end{gathered}\]	\(a\): a coefficient in the polynomial or quadratic	Shortcut division by a linear divisor.

9. Quadratic Equations & Functions (14 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
130	Standard form of a quadratic	\[y = ax^{2} + bx + c\]	\(c\): c coefficient in the polynomial or quadratic \(y\): dependent variable, output, or y-coordinate	Write a quadratic by coefficients.
131	Vertex form of a quadratic	\[y = a(x - h)^{2} + k\]	\(a\): a coefficient in the polynomial or quadratic \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Write a quadratic by vertex and stretch.
132	Factored form of a quadratic	\[y = a(x - r1)(x - r2)\]	\(r_1\): root or solution of the quadratic \(r_2\): root or solution of the quadratic \(a\): a coefficient in the polynomial or quadratic \(r\): radius, rate, ratio, remainder, or root \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Write a quadratic by roots.
133	Quadratic formula	\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(x\): unknown, input variable, or x-coordinate	Solve ax^2 + bx + c = 0.
134	Discriminant formula	\[D=b^2-4ac\]	\(D\): discriminant \(b\): b coefficient in the polynomial or quadratic	Determine the type and number of roots.
135	Axis of symmetry formula	\[x=-\frac{b}{2a}\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(x\): unknown, input variable, or x-coordinate	Find the vertical symmetry line.
136	Vertex formula	\[h = -b/(2a), k = f(h)\]	\(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(f\): function f, frequency, or matrix entry \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes	Find the vertex from standard form.
137	Completing the square formula	\[x^2+bx=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2\]	\(b\): b coefficient in the polynomial or quadratic \(x\): unknown, input variable, or x-coordinate	Rewrite a quadratic as a square plus constant.
138	Sum of roots formula	\[r1 + r2 = -b/a\]	\(r_1\): root or solution of the quadratic \(r_2\): root or solution of the quadratic \(a\): a coefficient in the polynomial or quadratic \(b\): b coefficient in the polynomial or quadratic \(r\): radius, rate, ratio, remainder, or root	Use Vieta relations for a quadratic.
139	Product of roots formula	\[r1 r2 = c/a\]	\(r_1\): root or solution of the quadratic \(r_2\): root or solution of the quadratic \(a\): a coefficient in the polynomial or quadratic \(c\): c coefficient in the polynomial or quadratic \(r\): radius, rate, ratio, remainder, or root	Use Vieta relations for a quadratic.
140	Maximum value of a quadratic formula	\[\begin{gathered}\text{max = k when a < 0 in y = a(x - h)\^{}2 + k}\end{gathered}\]	\(a\): a coefficient in the polynomial or quadratic \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find peak value of a downward-opening parabola.
141	Minimum value of a quadratic formula	\[\begin{gathered}\text{min = k when a > 0 in y = a(x - h)\^{}2 + k}\end{gathered}\]	\(a\): a coefficient in the polynomial or quadratic \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find lowest value of an upward-opening parabola.
142	Parabola opening rule	\[\begin{gathered}\text{a > 0 opens up}\\\text{a < 0 opens down}\end{gathered}\]	\(a\): a coefficient in the polynomial or quadratic	Read direction from the leading coefficient.
143	Quadratic regression formula	\[y = ax^{2} + bx + c\]	\(c\): c coefficient in the polynomial or quadratic \(y\): dependent variable, output, or y-coordinate	Fit a quadratic model to data.

10. Functions (16 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
144	Function notation formula	\[y = f(x)\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Name an output rule by its input.
145	Domain rule	\[\begin{gathered}\text{domain = all allowed input values x}\end{gathered}\]	\(x\): unknown, input variable, or x-coordinate	State the inputs for which a function is defined.
146	Range rule	\[\begin{gathered}\text{range = \{f(x): x in domain\}}\end{gathered}\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	State all possible outputs.
147	Composite function formula	\[(f\circ g)(x)=f(g(x))\]	\(f\): function f, frequency, or matrix entry \(g\): function g, greatest common factor, or gravitational/constant term \(x\): unknown, input variable, or x-coordinate	Apply one function inside another.
148	Inverse function formula	\[f^(-1)(f(x)) = x\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Reverse a one-to-one function.
149	Even function rule	\[\begin{gathered}\text{f(-x) = f(x)}\end{gathered}\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Test y-axis symmetry.
150	Odd function rule	\[\begin{gathered}\text{f(-x) = -f(x)}\end{gathered}\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Test origin symmetry.
151	Piecewise function formula	\[\begin{gathered}\text{f(x) = case rule by interval}\end{gathered}\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Use different rules on different domains.
152	Function transformation formulas	\[\begin{gathered}\text{y = a f(b(x - h)) + k}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Combine shifts, reflections, stretches, and compressions.
153	Horizontal shift formula	\[y = f(x - h)\]	\(f\): function f, frequency, or matrix entry \(h\): height, step size, or half-life interval \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Move a graph right h units when h is positive.
154	Vertical shift formula	\[y = f(x) + k\]	\(f\): function f, frequency, or matrix entry \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Move a graph up k units when k is positive.
155	Reflection formula	\[\begin{gathered}\text{y = -f(x) reflects over x-axis}\\\text{y = f(-x) reflects over y-axis}\end{gathered}\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Flip a graph across an axis.
156	Stretch formula	\[\begin{gathered}\text{y = a f(x), \|a\| > 1}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Stretch outputs vertically.
157	Compression formula	\[\begin{gathered}\text{y = a f(x), 0 < \|a\| < 1}\end{gathered}\]	\(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Compress outputs vertically.
158	Average rate of change formula	\[\frac{f(b)-f(a)}{b-a}\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry	Find average change over an interval.
159	Difference quotient formula	\[\frac{f(x+h)-f(x)}{h}\]	\(f\): function f, frequency, or matrix entry \(h\): height, step size, or half-life interval \(x\): unknown, input variable, or x-coordinate	Measure a function change over input step h.

11. Exponential & Logarithmic Functions (15 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
160	Exponential growth formula	\[y = a(1 + r)^{t}\]	\(a\): a base, initial value, or constant \(r\): radius, rate, ratio, remainder, or root \(t\): time, parameter, or integration variable \(y\): dependent variable, output, or y-coordinate	Model growth by a fixed percent rate.
161	Exponential decay formula	\[y = a(1 - r)^{t}\]	\(a\): a base, initial value, or constant \(r\): radius, rate, ratio, remainder, or root \(t\): time, parameter, or integration variable \(y\): dependent variable, output, or y-coordinate	Model decay by a fixed percent rate.
162	General exponential function formula	\[y = ab^{x}\]	\(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Model repeated multiplication by base b.
163	Compound interest formula	\[A=P\left(1+\frac{r}{n}\right)^{nt}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(P\): principal amount \(n\): number of terms, sides, observations, trials, or periods \(r\): interest rate per period	Calculate interest compounded n times per year.
164	Continuously compounded interest formula	\[A=Pe^{rt}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row	Calculate interest compounded continuously.
165	Half-life formula	\[A = A0(1/2)^(t/h)\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(h\): height, step size, or half-life interval \(t\): time, parameter, or integration variable	Model decay by repeated halving.
166	Doubling time formula	\[T = \ln(2) / k for y = ae^(kt)\]	\(T\): time, trapezoidal estimate, or transformation T \(k\): constant, scale factor, index, or number of successes \(y\): dependent variable, output, or y-coordinate	Find time needed to double under continuous growth.
167	Logarithmic form formula	\[y = log_b(x)\]	\(b\): b base, initial value, or constant \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Write the exponent needed to make x from base b.
168	Exponential form formula	\[b^y=x\Longleftrightarrow y=\log_b(x)\]	\(b\): b base, initial value, or constant \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Convert between log and exponential form.
169	Product rule of logarithms	\[log_b(MN) = log_b(M) + log_b(N)\]	\(M\): midpoint, matrix, mass, or midpoint sum \(N\): total count, population size, or original number \(b\): b base, initial value, or constant	Expand the log of a product.
170	Quotient rule of logarithms	\[log_b(M/N) = log_b(M) - log_b(N)\]	\(M\): midpoint, matrix, mass, or midpoint sum \(N\): total count, population size, or original number \(b\): b base, initial value, or constant	Expand the log of a quotient.
171	Power rule of logarithms	\[log_b(M^{p}) = p log_b(M)\]	\(M\): midpoint, matrix, mass, or midpoint sum \(b\): b base, initial value, or constant \(p\): rate, probability, or focus distance	Move an exponent out of a logarithm.
172	Change of base formula	\[\log_b(x)=\frac{\ln x}{\ln b}\]	\(b\): b base, initial value, or constant \(x\): unknown, input variable, or x-coordinate	Evaluate logs with a different base.
173	Natural logarithm formula	\[\ln(x) = log_e(x)\]	\(e\): Euler number for natural exponential growth, or a matrix entry \(x\): unknown, input variable, or x-coordinate	Use base e logarithms.
174	Common logarithm formula	\[\log(x) = log_10(x)\]	\(x\): unknown, input variable, or x-coordinate	Use base 10 logarithms.

12. Sequences & Series (11 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
175	Arithmetic sequence nth-term formula	\[a_n=a_1+(n-1)d\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(a\): a known number, coefficient, side length, or constant \(d\): common difference \(n\): number of terms, sides, observations, trials, or periods	Find a term in an arithmetic sequence.
176	Arithmetic series sum formula	\[S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)\]	\(a_1\): first term of a sequence \(S\): sum, series total, sale price, or named set S \(a\): a known number, coefficient, side length, or constant \(d\): common difference \(n\): number of terms, sides, observations, trials, or periods	Sum the first n arithmetic terms.
177	Geometric sequence nth-term formula	\[a_n=a_1r^{n-1}\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods \(r\): common ratio	Find a term in a geometric sequence.
178	Geometric series sum formula	\[S_n=\frac{a_1(1-r^n)}{1-r},\quad r\ne1\]	\(a_1\): first term of a sequence \(S\): sum, series total, sale price, or named set S \(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods \(r\): common ratio	Sum the first n geometric terms.
179	Infinite geometric series formula	\[S=\frac{a_1}{1-r},\quad \|r\|<1\]	\(a_1\): first term of a sequence \(S\): sum, series total, sale price, or named set S \(a\): a known number, coefficient, side length, or constant \(r\): common ratio	Sum an infinite geometric series that converges.
180	Recursive sequence formula	\[\begin{gathered}\text{a\_n = f(a\_(n-1)), with starting value a\_1}\end{gathered}\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(n\): number of terms, sides, observations, trials, or periods	Define each term from earlier term(s).
181	Fibonacci sequence formula	\[F_n = F_(n-1) + F_(n-2), F_1 = 1, F_2 = 1\]	\(F\): function, force, or named coefficient F \(n\): number of terms, sides, observations, trials, or periods	Generate Fibonacci terms recursively.
182	Sigma notation formula	\[\sum_{i=m}^{n} a_i\]	\(a_i\): ith term or indexed value \(a\): a known number, coefficient, side length, or constant \(i\): index value or matrix row index \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Write repeated addition compactly.
183	Partial sum formula	\[S_n = \sum_{i=1}^{n} a_i\]	\(a_i\): ith term or indexed value \(S\): sum, series total, sale price, or named set S \(a\): a known number, coefficient, side length, or constant \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods	Add the first n terms of a sequence.
184	Finite series formula	\[S_n = a_1 + a_2 + ... + a_n\]	\(a_n\): nth term of a sequence \(a_1\): first term of a sequence \(S\): sum, series total, sale price, or named set S \(a\): a known number, coefficient, side length, or constant \(n\): number of terms, sides, observations, trials, or periods	Represent a series with finitely many terms.
185	Infinite series formula	\[S = \lim(n \to \infty) S_n\]	\(S\): sum, series total, sale price, or named set S \(n\): number of terms, sides, observations, trials, or periods	Define an infinite series by a limit of partial sums.

13. Coordinate Geometry (15 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
186	Distance formula	\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(d\): distance, diameter, divisor, or common difference \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Find distance between two points.
187	Midpoint formula	\[M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(M\): midpoint, matrix, mass, or midpoint sum \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Find the point halfway between two points.
188	Section formula	\[P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))\]	\(P\): perimeter \(m\): mass, slope, index, or constant \(n\): number of terms, sides, observations, trials, or periods	Find a point dividing a segment in ratio m:n.
189	Slope formula	\[m = (y2 - y1)/(x2 - x1)\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(m\): slope \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Find steepness between two points.
190	Equation of a line	\[y - y1 = m(x - x1)\]	\(x_1\): x-1 coordinate of a point \(y_1\): y-1 coordinate of a point \(m\): mass, slope, index, or constant \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Write a line through a point with slope m.
191	Equation of a circle	\[(x - h)^{2} + (y - k)^{2} = r^{2}\]	\(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(r\): radius \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Write a circle by center and radius.
192	Center-radius form of a circle	\[(x - h)^{2} + (y - k)^{2} = r^{2}\]	\(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(r\): radius \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Read center (h,k) and radius r.
193	General form of a circle	\[x^{2} + y^{2} + Dx + Ey + F = 0\]	\(F\): function, force, or named coefficient F \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Write a circle after expansion.
194	Equation of a parabola	\[(x - h)^{2} = 4p(y - k) or (y - k)^{2} = 4p(x - h)\]	\(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(p\): rate, probability, or focus distance \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Represent a parabola by vertex and focus distance.
195	Equation of an ellipse	\[(x - h)^{2}/a^{2} + (y - k)^{2}/b^{2} = 1\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Represent an axis-aligned ellipse.
196	Equation of a hyperbola	\[(x - h)^{2}/a^{2} - (y - k)^{2}/b^{2} = 1\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Represent a horizontal axis-aligned hyperbola.
197	Translation formula	\[(x, y) \to (x + a, y + b)\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Move a point by vector (a,b).
198	Reflection formula	\[\begin{gathered}\text{over x-axis: (x,y)->(x,-y)}\\\text{over y-axis: (x,y)->(-x,y)}\end{gathered}\]	\(x\): x-coordinate or input value \(y\): y-coordinate or output value	Reflect a point across an axis.
199	Rotation formula	\[\begin{gathered}\text{x' = x cos theta - y sin theta}\\\text{y' = x sin theta + y cos theta}\end{gathered}\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(x\): x-coordinate or input value \(y\): y-coordinate or output value	Rotate a point about the origin.
200	Dilation formula	\[(x, y) \to (kx, ky)\]	\(x\): x-coordinate or input value \(y\): y-coordinate or output value	Scale a point from the origin by factor k.

14. Basic Geometry (16 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
201	Point formula	\[\begin{gathered}\text{point = (x, y)}\end{gathered}\]	\(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Represent a location with coordinates.
202	Line formula	\[\begin{gathered}\text{Ax + By + C = 0}\end{gathered}\]	\(C\): constant, coefficient, circumference, or third event as used in the row	Represent a straight line in general form.
203	Ray formula	\[\begin{gathered}\text{R(t) = P + tv, t >= 0}\end{gathered}\]	\(P\): perimeter \(R\): radius, right sum, range endpoint, or outer radius \(t\): time, parameter, or integration variable	Represent a ray from endpoint P in direction v.
204	Line segment formula	\[length = \sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}\]	\(x_1\): x-1 coordinate of a point \(x_2\): x-2 coordinate of a point \(y_1\): y-1 coordinate of a point \(y_2\): y-2 coordinate of a point \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find the length of a segment.
205	Angle sum formula	\[\begin{gathered}\text{angles on a line = 180°}\\\text{angles around a point = 360°}\end{gathered}\]	\(a\): a side length or coordinate constant	Use standard angle totals.
206	Complementary angles formula	\[A + B = 90°\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(B\): angle, base area, or set B as used in the row	Relate two angles that make a right angle.
207	Supplementary angles formula	\[A + B = 180°\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(B\): angle, base area, or set B as used in the row	Relate two angles that make a straight angle.
208	Vertical angles rule	\[\begin{gathered}\text{vertical angles are equal}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Use opposite angles formed by intersecting lines.
209	Linear pair rule	\[\begin{gathered}\text{adjacent linear-pair angles sum to 180°}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Use angles sharing a straight line.
210	Triangle angle sum formula	\[A+B+C=180^\circ\]	\(A\): area or angle A, depending on the row \(B\): angle, base area, or set B as used in the row \(C\): angle C	Find missing angles in triangles.
211	Exterior angle theorem	\[\begin{gathered}\text{exterior angle = sum of two remote interior angles}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Relate a triangle exterior angle to interior angles.
212	Polygon interior angle sum formula	\[\text{sum}=(n-2)180^\circ\]	\(n\): number of terms, sides, observations, trials, or periods	Find total interior angle measure.
213	Polygon exterior angle sum formula	\[\sum = 360°\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find total exterior angle measure for a convex polygon.
214	Regular polygon interior angle formula	\[\text{each interior angle}=\frac{(n-2)180^\circ}{n}\]	\(n\): number of terms, sides, observations, trials, or periods	Find one angle in a regular polygon.
215	Regular polygon exterior angle formula	\[each exterior angle = 360°/n\]	\(n\): number of terms, sides, observations, trials, or periods	Find one exterior angle in a regular polygon.
216	Diagonal formula for polygons	\[\text{diagonals}=\frac{n(n-3)}{2}\]	\(n\): number of terms, sides, observations, trials, or periods	Count diagonals in an n-sided polygon.

15. Triangles (16 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
217	Triangle perimeter formula	\[P = a + b + c\]	\(P\): perimeter \(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant	Add all three side lengths.
218	Triangle area formula	\[A=\frac{1}{2}bh\]	\(A\): area or angle A, depending on the row	Find area from base and height.
219	Right triangle area formula	\[A = (1/2)(leg1)(leg2)\]	\(A\): area or angle A, depending on the row	Use perpendicular legs as base and height.
220	Equilateral triangle area formula	\[A=\frac{\sqrt{3}}{4}s^2\]	\(A\): area or angle A, depending on the row \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find area from side length s.
221	Heron's formula	\[A=\sqrt{s(s-a)(s-b)(s-c)},\quad s=\frac{a+b+c}{2}\]	\(A\): area or angle A, depending on the row \(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find triangle area from three sides.
222	Pythagorean theorem	\[a^2+b^2=c^2\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant	Relate legs and hypotenuse in a right triangle.
223	Converse of Pythagorean theorem	\[\begin{gathered}\text{if a\^{}2 + b\^{}2 = c\^{}2, triangle is right}\end{gathered}\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant	Test whether side lengths form a right triangle.
224	Special right triangle 45-45-90 formulas	\[x,\ x,\ x\sqrt{2}\]	\(x\): unknown, input variable, or x-coordinate	Use side ratios for an isosceles right triangle.
225	Special right triangle 30-60-90 formulas	\[x,\ x\sqrt{3},\ 2x\]	\(x\): unknown, input variable, or x-coordinate	Use side ratios opposite 30°, 60°, and 90°.
226	Triangle inequality theorem	\[a + b > c, a + c > b, b + c > a\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant	Check whether three sides can form a triangle.
227	Similar triangles proportion formula	\[\begin{gathered}\text{a/b = c/d = scale factor}\end{gathered}\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant \(d\): distance, diameter, divisor, or common difference	Relate corresponding sides in similar triangles.
228	Congruent triangles rules	\[\begin{gathered}\text{SSS, SAS, ASA, AAS, HL}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Prove triangles have matching size and shape.
229	Median formula	\[m_a = (1/2)\sqrt{2b^{2} + 2c^{2} - a^{2}}\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant \(m\): mass, slope, index, or constant	Find the median to side a.
230	Altitude formula	\[h = 2A / b\]	\(A\): area or angle A, depending on the row \(b\): b side length or coordinate constant \(h\): height, step size, or half-life interval	Find height from area and base.
231	Angle bisector theorem	\[BD/DC = AB/AC\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Relate side split by an angle bisector.
232	Perpendicular bisector theorem	\[\begin{gathered}\text{if P is on perpendicular bisector of AB, PA = PB}\end{gathered}\]	\(P\): perimeter	Use equal distances from segment endpoints.

16. Quadrilaterals (16 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
233	Square perimeter formula	\[P = 4s\]	\(P\): perimeter \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find square perimeter from side length.
234	Square area formula	\[A = s^{2}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find square area from side length.
235	Rectangle perimeter formula	\[P=2(l+w)\]	\(P\): perimeter \(l\): length or slant height \(w\): width, weight, or vector component	Find rectangle perimeter.
236	Rectangle area formula	\[A=lw\]	\(A\): area, amount, angle, event, or matrix named A as used in the row	Find rectangle area.
237	Parallelogram perimeter formula	\[P = 2(a + b)\]	\(P\): perimeter \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Find perimeter from adjacent side lengths.
238	Parallelogram area formula	\[A = bh\]	\(A\): area, amount, angle, event, or matrix named A as used in the row	Find area from base and height.
239	Rhombus perimeter formula	\[P = 4s\]	\(P\): perimeter \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find rhombus perimeter.
240	Rhombus area formula	\[A=\frac{d_1d_2}{2}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(d\): distance, diameter, divisor, or common difference	Find rhombus area from diagonals.
241	Trapezoid area formula	\[A=\frac{1}{2}(b_1+b_2)h\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(b\): b known number, coefficient, side length, or constant \(h\): height, step size, or half-life interval	Find trapezoid area from bases and height.
242	Kite area formula	\[A = (d1 d2)/2\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(d\): distance, diameter, divisor, or common difference	Find kite area from diagonals.
243	Diagonal formula of rectangle	\[d = \sqrt{l^{2} + w^{2}}\]	\(d\): distance, diameter, divisor, or common difference \(l\): length or slant height \(w\): width, weight, or vector component	Find a rectangle diagonal.
244	Diagonal formula of square	\[d = s \sqrt{2}\]	\(d\): distance, diameter, divisor, or common difference \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find a square diagonal.
245	Properties of parallelograms formulas	\[\begin{gathered}\text{opposite sides equal}\\\text{opposite angles equal}\\\text{diagonals bisect}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Use core parallelogram relationships.
246	Properties of rectangles formulas	\[\begin{gathered}\text{all angles = 90°}\\\text{d\^{}2 = l\^{}2 + w\^{}2}\end{gathered}\]	\(d\): distance, diameter, divisor, or common difference \(l\): length or slant height \(w\): width, weight, or vector component	Use rectangle angle and diagonal facts.
247	Properties of rhombuses formulas	\[\begin{gathered}\text{all sides equal}\\\text{diagonals are perpendicular}\\\text{A = d1d2/2}\end{gathered}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(d\): distance, diameter, divisor, or common difference	Use rhombus side and diagonal facts.
248	Properties of squares formulas	\[\begin{gathered}\text{P = 4s}\\\text{A = s\^{}2}\\\text{d = s sqrt(2)}\end{gathered}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(P\): perimeter \(d\): distance, diameter, divisor, or common difference \(s\): semiperimeter, arc length, side length, or sample standard deviation	Use square side, area, and diagonal facts.

17. Circles (14 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
249	Circle circumference formula	\[C=2\pi r=\pi d\]	\(\pi\): the constant pi, approximately 3.14159 \(C\): circumference \(d\): diameter \(r\): radius	Find distance around a circle.
250	Circle area formula	\[A=\pi r^2\]	\(\pi\): the constant pi, approximately 3.14159 \(A\): area, amount, angle, event, or matrix named A as used in the row \(r\): radius	Find area inside a circle.
251	Radius formula	\[r = d/2 = C/(2pi)\]	\(\pi\): the constant pi, approximately 3.14159 \(C\): circumference \(d\): diameter \(r\): radius	Find radius from diameter or circumference.
252	Diameter formula	\[d = 2r = C/\pi\]	\(\pi\): the constant pi, approximately 3.14159 \(C\): circumference \(d\): diameter \(r\): radius	Find diameter from radius or circumference.
253	Arc length formula	\[s=r\theta\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(r\): radius \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find arc length when theta is in radians.
254	Sector area formula	\[A=\frac{1}{2}r^2\theta\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(A\): area, amount, angle, event, or matrix named A as used in the row \(r\): radius	Find sector area when theta is in radians.
255	Segment area formula	\[A = (r^{2}/2)(\theta - \sin \theta)\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(A\): area, amount, angle, event, or matrix named A as used in the row \(r\): radius	Find area between chord and arc.
256	Central angle formula	\[\theta = s/r\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(r\): radius \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find central angle from arc length and radius.
257	Inscribed angle formula	\[inscribed angle = (1/2)(intercepted arc)\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Relate an inscribed angle to its arc.
258	Chord length formula	\[chord = 2r \sin(\theta/2)\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(r\): radius	Find chord length from radius and central angle.
259	Tangent length formula	\[PA = PB\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Use equal tangent lengths from the same external point.
260	Secant-tangent theorem	\[\text{tangent}^2=(\text{external secant})(\text{whole secant})\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Solve tangent and secant lengths.
261	Intersecting chords theorem	\[a\times b=c\times d\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): diameter	Use products of chord segments inside a circle.
262	Circle equation formula	\[(x - h)^{2} + (y - k)^{2} = r^{2}\]	\(h\): height, step size, or half-life interval \(k\): constant, scale factor, index, or number of successes \(r\): radius \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Graph a circle in the coordinate plane.

18. Three-Dimensional Geometry (19 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
263	Cube volume formula	\[V = s^{3}\]	\(V\): volume, value, or vector V \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find volume from cube side length.
264	Cube surface area formula	\[SA = 6s^{2}\]	\(SA\): surface area \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find total area of six square faces.
265	Rectangular prism volume formula	\[V = lwh\]	\(V\): volume, value, or vector V	Find box volume.
266	Rectangular prism surface area formula	\[SA = 2(lw + lh + wh)\]	\(SA\): surface area	Find total area of a rectangular prism.
267	Triangular prism volume formula	\[V=(\text{area of triangular base})\times\text{length}\]	\(V\): volume, value, or vector V	Find volume of a triangular prism.
268	Prism surface area formula	\[SA = 2B + Ph\]	\(SA\): surface area \(B\): angle, base area, or set B as used in the row	Find surface area from base area B and base perimeter P.
269	Cylinder volume formula	\[V=\pi r^2h\]	\(\pi\): the constant pi, approximately 3.14159 \(V\): volume, value, or vector V \(h\): height, step size, or half-life interval \(r\): radius	Find cylinder volume.
270	Cylinder surface area formula	\[SA = 2pi r^{2} + 2pi rh\]	\(SA\): surface area \(\pi\): the constant pi, approximately 3.14159 \(r\): radius	Find total cylinder surface area.
271	Cone volume formula	\[V = (1/3)\pi r^{2} h\]	\(\pi\): the constant pi, approximately 3.14159 \(V\): volume, value, or vector V \(h\): height, step size, or half-life interval \(r\): radius	Find cone volume.
272	Cone surface area formula	\[SA = \pi r^{2} + \pi r l\]	\(SA\): surface area \(\pi\): the constant pi, approximately 3.14159 \(l\): length or slant height \(r\): radius	Find total cone surface area with slant height l.
273	Sphere volume formula	\[V=\frac{4}{3}\pi r^3\]	\(\pi\): the constant pi, approximately 3.14159 \(V\): volume, value, or vector V \(r\): radius	Find sphere volume.
274	Sphere surface area formula	\[SA = 4pi r^{2}\]	\(SA\): surface area \(\pi\): the constant pi, approximately 3.14159 \(r\): radius	Find sphere surface area.
275	Hemisphere volume formula	\[V = (2/3)\pi r^{3}\]	\(\pi\): the constant pi, approximately 3.14159 \(V\): volume, value, or vector V \(r\): radius	Find half-sphere volume.
276	Hemisphere surface area formula	\[\begin{gathered}\text{SA = 3pi r\^{}2 total}\\\text{curved area = 2pi r\^{}2}\end{gathered}\]	\(SA\): surface area \(\pi\): the constant pi, approximately 3.14159 \(r\): radius	Find total or curved hemisphere surface area.
277	Pyramid volume formula	\[V = (1/3)Bh\]	\(V\): volume, value, or vector V	Find pyramid volume from base area and height.
278	Pyramid surface area formula	\[SA = B + (1/2)Pl\]	\(SA\): surface area \(B\): angle, base area, or set B as used in the row	Find pyramid surface area with slant height l.
279	Frustum volume formula	\[V = (h/3)(A1 + A2 + \sqrt{A1A2})\]	\(A_1\): area of one of the parallel bases \(A_2\): area of one of the parallel bases \(A\): area, amount, angle, event, or matrix named A as used in the row \(V\): volume, value, or vector V \(h\): height, step size, or half-life interval	Find volume between two similar parallel bases.
280	Composite solid volume formula	\[V_total = \sum V_parts - \sum V_removed\]	\(V\): volume, value, or vector V	Combine volumes of multiple solids.
281	Density formula	\[\rho = m / V\]	\(\rho\): density \(V\): volume, value, or vector V \(m\): mass, slope, index, or constant	Relate mass and volume.

19. Measurement (12 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
282	Length conversion formulas	\[\begin{gathered}\text{new length = old length x conversion factor}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Convert between length units.
283	Area conversion formulas	\[\begin{gathered}\text{new area = old area x (linear factor)\^{}2}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Convert between area units.
284	Volume conversion formulas	\[\begin{gathered}\text{new volume = old volume x (linear factor)\^{}3}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Convert between volume units.
285	Mass conversion formulas	\[\begin{gathered}\text{new mass = old mass x conversion factor}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Convert between mass units.
286	Time conversion formulas	\[\begin{gathered}\text{new time = old time x conversion factor}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Convert seconds, minutes, hours, days, and years.
287	Temperature conversion formulas	\[F = (9/5)C + 32; C = (5/9)(F - 32); K = C + 273.15\]	\(C\): constant, coefficient, circumference, or third event as used in the row \(F\): function, force, or named coefficient F	Convert Celsius, Fahrenheit, and Kelvin.
288	Speed formula	\[speed = distance / time\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure how fast distance is covered.
289	Distance formula	\[\begin{gathered}\text{distance = rate x time}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find distance from rate and time.
290	Time formula	\[\begin{gathered}\text{time = distance / rate}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find time from distance and rate.
291	Rate formula	\[\begin{gathered}\text{rate = quantity / time}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find change per unit time.
292	Unit conversion formula	\[\begin{gathered}\text{converted value = original value x unit ratio}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Convert units using a factor equal to 1.
293	Dimensional analysis formula	\[\begin{gathered}\text{value x (desired unit / given unit)}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Cancel units to check a conversion path.

20. Trigonometry (20 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
294	Sine ratio formula	\[\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\]	\(\theta\): angle measure, usually in radians unless degrees are stated	Find sine in a right triangle.
295	Cosine ratio formula	\[\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\]	\(\theta\): angle measure, usually in radians unless degrees are stated	Find cosine in a right triangle.
296	Tangent ratio formula	\[\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\]	\(\theta\): angle measure, usually in radians unless degrees are stated	Find tangent in a right triangle.
297	Reciprocal trigonometric ratios	\[\begin{gathered}\text{csc theta = 1/sin theta}\\\text{sec theta = 1/cos theta}\\\text{cot theta = 1/tan theta}\end{gathered}\]	\(\theta\): angle measure, usually in radians unless degrees are stated	Use reciprocal trig functions.
298	Pythagorean trigonometric identity	\[\sin^2\theta+\cos^2\theta=1\]	\(\theta\): angle measure, usually in radians unless degrees are stated	Simplify trig expressions.
299	Complementary angle identities	\[\begin{gathered}\text{sin(90° - theta) = cos theta}\\\text{tan(90° - theta) = cot theta}\end{gathered}\]	\(\theta\): angle measure, usually in radians unless degrees are stated	Relate cofunctions of complementary angles.
300	Unit circle formulas	\[(x, y) = (\cos \theta, \sin \theta)\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Read sine and cosine from a unit-circle point.
301	Radian measure formula	\[\theta = s/r\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(r\): radius, rate, ratio, remainder, or root \(s\): semiperimeter, arc length, side length, or sample standard deviation	Define radians from arc length and radius.
302	Degree-radian conversion formula	\[\text{radians}=\text{degrees}\times\frac{\pi}{180}\]	\(\pi\): the constant pi, approximately 3.14159	Convert angle units.
303	Arc length using radians formula	\[s = r \theta\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(r\): radius, rate, ratio, remainder, or root \(s\): semiperimeter, arc length, side length, or sample standard deviation	Find arc length with theta in radians.
304	Sector area using radians formula	\[A = (1/2)r^{2} \theta\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(A\): area or angle A, depending on the row \(r\): radius, rate, ratio, remainder, or root	Find sector area with theta in radians.
305	Law of sines	\[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\]	\(A\): area or angle A, depending on the row \(B\): angle, base area, or set B as used in the row \(C\): angle C \(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant	Solve non-right triangles with side-angle pairs.
306	Law of cosines	\[c^2=a^2+b^2-2ab\cos C\]	\(C\): angle C \(a\): a side length or coordinate constant \(b\): b side length or coordinate constant \(c\): c side length or coordinate constant	Solve non-right triangles with included angle or three sides.
307	Area of triangle using sine formula	\[A = (1/2)ab \sin C\]	\(A\): area or angle A, depending on the row \(C\): angle C	Find triangle area from two sides and included angle.
308	Sum identities	\[\sin(a+b)=\sin a \cos b + \cos a \sin b; \cos(a+b)=\cos a \cos b - \sin a \sin b\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant	Expand trig functions of sums.
309	Difference identities	\[\sin(a-b)=\sin a \cos b - \cos a \sin b; \cos(a-b)=\cos a \cos b + \sin a \sin b\]	\(a\): a side length or coordinate constant \(b\): b side length or coordinate constant	Expand trig functions of differences.
310	Double-angle identities	\[\sin 2x = 2sin x \cos x; \cos 2x = \cos^{2} x - \sin^{2} x\]	\(x\): unknown, input variable, or x-coordinate	Simplify trig expressions with twice an angle.
311	Half-angle identities	\[\sin^{2}(x/2) = (1 - \cos x)/2; \cos^{2}(x/2) = (1 + \cos x)/2\]	\(x\): unknown, input variable, or x-coordinate	Work with half-angle expressions.
312	Inverse trigonometric formulas	\[\theta = \sin^(-1)(x), \cos^(-1)(x), or \tan^(-1)(x)\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(x\): unknown, input variable, or x-coordinate	Find an angle from a trig ratio.
313	Trigonometric equation formulas	\[\sin x = a \to x = \arcsin(a) + 2pi k or \pi - \arcsin(a) + 2pi k\]	\(\pi\): the constant pi, approximately 3.14159 \(a\): a side length or coordinate constant \(k\): constant, scale factor, index, or number of successes	Write general solutions for trig equations.

21. Data, Statistics & Probability (33 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
314	Mean formula	\[\bar{x}=\frac{\sum x_i}{n}\]	\(x_i\): ith data value or sample value \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Find the arithmetic average.
315	Median formula	\[\begin{gathered}\text{median = middle ordered value}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find the center of ordered data.
316	Mode formula	\[\begin{gathered}\text{mode = most frequent value}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find the most repeated value.
317	Range formula	\[range = maximum - minimum\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure spread from smallest to largest.
318	Interquartile range formula	\[\operatorname{IQR}=Q_3-Q_1\]	\(IQR\): interquartile range \(Q\): quartile, matrix quotient, or named quantity Q	Measure spread of the middle half.
319	Quartile formula	\[Q_k position = k(n + 1)/4\]	\(Q\): quartile, matrix quotient, or named quantity Q \(k\): constant, scale factor, index, or number of successes \(n\): number of terms, sides, observations, trials, or periods	Estimate quartile positions in ordered data.
320	Percentile formula	\[\text{percentile rank}=\frac{\text{values below}}{n}\times100\]	\(n\): number of terms, sides, observations, trials, or periods	Measure relative standing in data.
321	Mean absolute deviation formula	\[MAD = (\sum \|x_i - mean\|) / n\]	\(MAD\): mean absolute deviation \(x_i\): ith data value or sample value \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Measure average absolute distance from the mean.
322	Variance formula	\[\begin{gathered}\text{variance = mean of squared deviations}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure average squared spread.
323	Standard deviation formula	\[\begin{gathered}\text{standard deviation = sqrt(variance)}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure typical distance from the mean.
324	Population variance formula	\[\sigma^2=\frac{\sum(x_i-\mu)^2}{N}\]	\(x_i\): ith data value or sample value \(\mu\): population mean \(\sigma\): population standard deviation \(N\): total count, population size, or original number \(i\): index value or matrix row index \(x\): unknown, input variable, or x-coordinate	Find variance for a full population.
325	Sample variance formula	\[s^2=\frac{\sum(x_i-\bar{x})^2}{n-1}\]	\(x_i\): ith data value or sample value \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(s\): semiperimeter, arc length, side length, or sample standard deviation \(x\): unknown, input variable, or x-coordinate	Estimate variance from a sample.
326	Population standard deviation formula	\[\sigma = \sqrt{(\sum (x_i - \mu)^{2}) / N}\]	\(x_i\): ith data value or sample value \(\mu\): population mean \(\sigma\): population standard deviation \(N\): total count, population size, or original number \(i\): index value or matrix row index \(x\): unknown, input variable, or x-coordinate	Find population standard deviation.
327	Sample standard deviation formula	\[s = \sqrt{(\sum (x_i - xbar)^{2}) / (n - 1)}\]	\(x_i\): ith data value or sample value \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(s\): semiperimeter, arc length, side length, or sample standard deviation \(x\): unknown, input variable, or x-coordinate	Find sample standard deviation.
328	z-score formula	\[z=\frac{x-\mu}{\sigma}\]	\(\mu\): population mean \(\sigma\): population standard deviation \(x\): unknown, input variable, or x-coordinate \(z\): z-score, z-coordinate, or standard normal value	Measure standard deviations from the mean.
329	Weighted mean formula	\[\bar{x}_w=\frac{\sum w_ix_i}{\sum w_i}\]	\(x_i\): ith data value or sample value \(w_i\): weight assigned to the ith value \(i\): index value or matrix row index \(w\): width, weight, or vector component \(x\): unknown, input variable, or x-coordinate	Average values with weights.
330	Frequency table formula	\[\begin{gathered}\text{N = sum f\_i}\end{gathered}\]	\(f_i\): frequency for the ith value or class \(N\): total count, population size, or original number \(f\): function f, frequency, or matrix entry \(i\): index value or matrix row index	Summarize counts by category or value.
331	Relative frequency formula	\[relative frequency = f / N\]	\(N\): total count, population size, or original number \(f\): function f, frequency, or matrix entry	Convert a count to a share.
332	Probability formula	\[P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes}}\]	\(E\): event \(P\): probability	Find basic probability.
333	Experimental probability formula	\[P(E) = successes / trials\]	\(E\): event \(P\): probability	Estimate probability from results.
334	Theoretical probability formula	\[\begin{gathered}\text{P(E) = favorable equally likely outcomes / all equally likely outcomes}\end{gathered}\]	\(E\): event \(P\): probability	Find probability from a model.
335	Complement probability formula	\[P(not E) = 1 - P(E)\]	\(E\): event \(P\): probability	Find the probability an event does not occur.
336	Addition rule of probability	\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]	\(A\): event or set A \(B\): event or set B \(P\): probability	Find probability of A or B.
337	Multiplication rule of probability	\[P(A \cap B) = P(A)P(B\|A)\]	\(A\): event or set A \(B\): event or set B \(P\): probability	Find probability of A and B.
338	Conditional probability formula	\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\]	\(A\): event or set A \(B\): event or set B \(P\): probability	Find probability of A given B.
339	Independent events formula	\[P(A \cap B) = P(A)P(B)\]	\(A\): event or set A \(B\): event or set B \(P\): probability	Test or use independence.
340	Dependent events formula	\[\begin{gathered}\text{P(A then B) = P(A)P(B\|A)}\end{gathered}\]	\(A\): event or set A \(B\): event or set B \(P\): probability	Handle probability when the first event changes the second.
341	Mutually exclusive events formula	\[P(A \cap B) = 0\]	\(A\): event or set A \(B\): event or set B \(P\): probability	Use when events cannot happen together.
342	Permutation formula	\[{}_nP_r=\frac{n!}{(n-r)!}\]	\(n\): number of terms, sides, observations, trials, or periods \(r\): radius, rate, ratio, remainder, or root	Count ordered selections.
343	Combination formula	\[{}_nC_r=\frac{n!}{r!(n-r)!}\]	\(n\): number of terms, sides, observations, trials, or periods \(r\): radius, rate, ratio, remainder, or root	Count unordered selections.
344	Binomial probability formula	\[P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\]	\(P\): probability \(X\): random variable, x-axis quantity, or set X \(k\): constant, scale factor, index, or number of successes \(n\): number of terms, sides, observations, trials, or periods \(p\): probability of success	Find exactly k successes in n independent trials.
345	Expected value formula	\[E(X)=\sum xP(x)\]	\(E\): event \(P\): probability \(X\): random variable, x-axis quantity, or set X \(x\): unknown, input variable, or x-coordinate	Find long-run average outcome.
346	Normal distribution formula	\[f(x)=1/(\sigma \sqrt{2pi}) e^(-((x-\mu)^{2})/(2sigma^{2}))\]	\(\mu\): population mean \(\sigma\): population standard deviation \(\pi\): the constant pi, approximately 3.14159 \(e\): Euler number for natural exponential growth, or a matrix entry \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Use the normal probability density curve.

22. Financial Mathematics (18 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
347	Simple interest formula	\[I=Prt\]	\(I\): interest amount	Calculate interest without compounding.
348	Compound interest formula	\[A=P\left(1+\frac{r}{n}\right)^{nt}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(P\): principal amount \(n\): number of terms, sides, observations, trials, or periods \(r\): interest rate per period	Calculate interest with periodic compounding.
349	Continuous compound interest formula	\[A=Pe^{rt}\]	\(A\): area, amount, angle, event, or matrix named A as used in the row	Calculate continuously compounded growth.
350	Future value formula	\[FV=PV(1+r)^n\]	\(PV\): present value \(FV\): future value \(n\): number of terms, sides, observations, trials, or periods \(r\): interest rate per period	Find future value after n periods.
351	Present value formula	\[PV=\frac{FV}{(1+r)^n}\]	\(PV\): present value \(FV\): future value \(n\): number of terms, sides, observations, trials, or periods \(r\): interest rate per period	Discount a future value to today.
352	Loan payment formula	\[PMT=\frac{Pr(1+r)^n}{(1+r)^n-1}\]	\(PMT\): fixed periodic payment \(P\): principal amount \(n\): number of terms, sides, observations, trials, or periods \(r\): interest rate per period	Find fixed payment for an amortizing loan.
353	Mortgage payment formula	\[M=\frac{Pr(1+r)^n}{(1+r)^n-1}\]	\(M\): mortgage payment \(P\): principal amount \(n\): number of terms, sides, observations, trials, or periods \(r\): interest rate per period	Find a fixed mortgage principal-and-interest payment.
354	Amortization formula	\[balance = P(1+r)^{k} - PMT((1+r)^{k} - 1)/r\]	\(PMT\): fixed periodic payment \(P\): principal amount \(k\): constant, scale factor, index, or number of successes \(r\): interest rate per period	Estimate remaining loan balance after k payments.
355	Depreciation formula	\[\begin{gathered}\text{straight-line depreciation = (cost - salvage) / useful life}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Spread asset cost evenly over time.
356	Markup formula	\[\text{markup}=\text{selling price}-\text{cost},\quad \text{markup \%}=\frac{\text{markup}}{\text{cost}}\times100\%\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure price increase over cost.
357	Markdown formula	\[\text{markdown}=\text{original price}-\text{sale price},\quad \text{markdown \%}=\frac{\text{markdown}}{\text{original}}\times100\%\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure price reduction from original price.
358	Discount formula	\[\begin{gathered}\text{sale price = original price(1 - discount rate)}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Apply a percentage discount.
359	Sales tax formula	\[\begin{gathered}\text{total = price(1 + tax rate)}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Add sales tax to a purchase.
360	Profit formula	\[\begin{gathered}\text{profit = revenue - cost}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure gain after costs.
361	Loss formula	\[\begin{gathered}\text{loss = cost - revenue}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Measure shortfall when cost exceeds revenue.
362	Commission formula	\[\text{commission}=\text{sales}\times\text{commission rate}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Calculate earnings from sales rate.
363	Break-even formula	\[\begin{gathered}\text{break-even units = fixed costs / (price - variable cost)}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find units needed to cover costs.
364	Inflation formula	\[\begin{gathered}\text{real value = nominal value / (1 + inflation rate)\^{}n}\end{gathered}\]	\(n\): number of terms, sides, observations, trials, or periods	Adjust money for price-level change.

23. Calculus Readiness / Precalculus (16 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
365	Limit notation formula	\[\lim(x \to a) f(x) = L\]	\(L\): length, left sum, or lower endpoint as used in the row \(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	State the value approached by a function.
366	Average rate of change formula	\[(f(b) - f(a)) / (b - a)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry	Find slope of a secant line.
367	Instantaneous rate of change formula	\[f'(a) = \lim(h \to 0) (f(a+h) - f(a))/h\]	\(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(h\): small input change or subinterval width	Define derivative at a point.
368	Difference quotient formula	\[(f(x+h) - f(x)) / h\]	\(f\): function f, frequency, or matrix entry \(h\): small input change or subinterval width \(x\): unknown, input variable, or x-coordinate	Prepare for derivative calculations.
369	Secant line slope formula	\[m_secant = (f(b) - f(a)) / (b - a)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(m\): slope	Find slope through two points on a curve.
370	Tangent line slope formula	\[m_tangent = f'(a)\]	\(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(m\): slope	Find slope of a curve at one point.
371	End behavior formulas	\[study f(x) as x \to \infty and x \to -\infty\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Describe graph behavior far left and right.
372	Polynomial end behavior rule	\[\begin{gathered}\text{leading term a\_n x\^{}n controls end behavior}\end{gathered}\]	\(a_n\): nth term of a sequence \(a\): a coefficient in the polynomial or quadratic \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Predict polynomial tails.
373	Rational function asymptote formulas	\[\begin{gathered}\text{vertical: denominator = 0}\\\text{horizontal/slant: compare degrees}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Analyze rational-function boundaries.
374	Horizontal asymptote formula	\[\begin{gathered}\text{deg top < deg bottom -> y=0}\\\text{equal degrees -> y=leading coefficient ratio}\end{gathered}\]	\(y\): dependent variable, output, or y-coordinate	Find horizontal asymptotes.
375	Vertical asymptote formula	\[\begin{gathered}\text{set simplified denominator = 0}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find x-values where a rational function is unbounded.
376	Slant asymptote formula	\[\begin{gathered}\text{slant asymptote = polynomial quotient when deg top = deg bottom + 1}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Find oblique asymptotes by division.
377	Rational function simplification formula	\[\begin{gathered}\text{factor numerator and denominator, then cancel common nonzero factors}\end{gathered}\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Simplify rational expressions and identify holes.
378	Trigonometric graph formulas	\[y = A \sin(B(x - C)) + D; period = 2pi/\|B\|\]	\(\pi\): the constant pi, approximately 3.14159 \(A\): area, amount, angle, event, or matrix named A as used in the row \(B\): second quantity, event, matrix, or coefficient named B \(C\): constant of integration \(D\): difference, determinant, diameter, or named constant D \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Graph sine and cosine transformations.
379	Exponential graph formulas	\[y = ab^(x - h) + k\]	\(h\): small input change or subinterval width \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Graph exponential transformations.
380	Logarithmic graph formulas	\[y = a log_b(x - h) + k\]	\(a\): a base, initial value, or constant \(b\): b base, initial value, or constant \(h\): small input change or subinterval width \(k\): constant, scale factor, index, or number of successes \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Graph logarithmic transformations.

24. Matrices & Vectors (18 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
381	Matrix addition formula	\[(A+B)_{ij}=A_{ij}+B_{ij}\]	\(A\): matrix or first event, depending on the row \(B\): second quantity, event, matrix, or coefficient named B	Add matching entries of same-size matrices.
382	Matrix subtraction formula	\[(A - B)_ij = A_ij - B_ij\]	\(A\): matrix or first event, depending on the row \(B\): second quantity, event, matrix, or coefficient named B	Subtract matching entries of same-size matrices.
383	Scalar multiplication formula	\[(cA)_ij = cA_ij\]	Notation: This row is a rule or named structure; use the named quantities and conditions exactly as stated.	Multiply every matrix entry by a scalar.
384	Matrix multiplication formula	\[(AB)_{ij}=\sum_k A_{ik}B_{kj}\]	\(A\): matrix or first event, depending on the row \(B\): second quantity, event, matrix, or coefficient named B \(k\): constant, scale factor, index, or number of successes	Multiply rows by columns.
385	Identity matrix formula	\[AI = IA = A\]	\(A\): matrix or first event, depending on the row	Use the matrix version of 1.
386	Zero matrix formula	\[A + 0 = A\]	\(A\): matrix or first event, depending on the row	Use the matrix version of 0.
387	Determinant of 2x2 matrix	\[\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Find determinant of a 2 by 2 matrix.
388	Determinant of 3x3 matrix	\[a(ei - fh) - b(di - fg) + c(dh - eg)\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant	Find determinant of [[a,b,c],[d,e,f],[g,h,i]].
389	Inverse of 2x2 matrix	\[A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\]	\(A\): matrix or first event, depending on the row \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Invert a 2 by 2 matrix when determinant is nonzero.
390	Matrix equation formula	\[Ax = b\]	\(b\): b known number, coefficient, side length, or constant	Represent linear equations compactly.
391	Vector magnitude formula	\[\|\|v\|\| = \sqrt{v1^{2} + v2^{2} + ... + vn^{2}}\]	\(v\): vector v, velocity, or value	Find vector length.
392	Vector addition formula	\[<a,b> + <c,d> = <a+c,b+d>\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Add vectors component by component.
393	Vector subtraction formula	\[<a,b> - <c,d> = <a-c,b-d>\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(c\): c known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference	Subtract vectors component by component.
394	Scalar multiplication of vector formula	\[k<a,b> = <ka,kb>\]	\(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(k\): constant, scale factor, index, or number of successes	Scale a vector.
395	Dot product formula	\[u \cdot v = \sum u_i v_i = \|\|u\|\|\|\|v\|\|\cos \theta\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(i\): index value or matrix row index \(u\): substitution variable or vector u \(v\): vector v, velocity, or value	Measure projection and angle relation.
396	Cross product formula	\[u\times v=\langle u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1\rangle\]	\(u\): substitution variable or vector u \(v\): vector v, velocity, or value	Find a 3D vector perpendicular to two vectors.
397	Unit vector formula	\[u_hat = u / \|\|u\|\|\]	\(u\): substitution variable or vector u	Convert a nonzero vector to length 1.
398	Direction angle formula	\[\theta = atan2(y, x)\]	\(\theta\): angle measure, usually in radians unless degrees are stated \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Find vector direction from components.

25. Logic, Sets & Discrete Math (15 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
399	Set union formula	\[A\cup B=\{x:x\in A\ \text{or}\ x\in B\}\]	\(A\): event or set A \(B\): event or set B \(x\): unknown, input variable, or x-coordinate	Combine elements from either set.
400	Set intersection formula	\[A\cap B=\{x:x\in A\ \text{and}\ x\in B\}\]	\(A\): event or set A \(B\): event or set B \(x\): unknown, input variable, or x-coordinate	Find shared elements.
401	Set complement formula	\[A'=\{x\in U:x\notin A\}\]	\(A\): event or set A \(x\): unknown, input variable, or x-coordinate	Find elements outside a set within universe U.
402	Difference of sets formula	\[A-B=\{x:x\in A\ \text{and}\ x\notin B\}\]	\(A\): event or set A \(B\): event or set B \(x\): unknown, input variable, or x-coordinate	Remove B elements from A.
403	Cartesian product formula	\[A\times B=\{(a,b):a\in A,\ b\in B\}\]	\(A\): event or set A \(B\): event or set B \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Build ordered pairs from two sets.
404	Venn diagram formula	\[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]	\(A\): event or set A \(B\): event or set B \(n\): number of terms, sides, observations, trials, or periods	Count elements in overlapping sets.
405	Inclusion-exclusion principle	\[n(A \cup B \cup C) = n(A)+n(B)+n(C)-n(AB)-n(AC)-n(BC)+n(ABC)\]	\(A\): event or set A \(B\): event or set B \(C\): constant, coefficient, circumference, or third event as used in the row \(n\): number of terms, sides, observations, trials, or periods	Count overlapping groups without double-counting.
406	Truth table formulas	\[\begin{gathered}\text{not p, p and q, p or q, p -> q}\end{gathered}\]	\(p\): rate, probability, or focus distance \(q\): quotient, probability of failure, or integer part	Evaluate compound logical statements.
407	Conditional statement formula	\[p \to q is equivalent to not p or q\]	\(p\): rate, probability, or focus distance \(q\): quotient, probability of failure, or integer part	Represent "if p, then q."
408	Converse formula	\[q \to p\]	\(p\): rate, probability, or focus distance \(q\): quotient, probability of failure, or integer part	Reverse a conditional statement.
409	Inverse formula	\[not p \to not q\]	\(p\): rate, probability, or focus distance \(q\): quotient, probability of failure, or integer part	Negate both parts of a conditional.
410	Contrapositive formula	\[not q \to not p\]	\(p\): rate, probability, or focus distance \(q\): quotient, probability of failure, or integer part	Use a logically equivalent form of p -> q.
411	Direct proof structure	\[\begin{gathered}\text{assume p, use definitions/theorems, conclude q}\end{gathered}\]	\(p\): rate, probability, or focus distance \(q\): quotient, probability of failure, or integer part Rule: Read the text condition before applying the symbols or calculator workflow.	Prove a conditional directly.
412	Indirect proof structure	\[\begin{gathered}\text{assume not q, derive contradiction, conclude q}\end{gathered}\]	\(q\): quotient, probability of failure, or integer part Rule: Read the text condition before applying the symbols or calculator workflow.	Prove by contradiction or contrapositive reasoning.
413	Mathematical induction formula	\[\begin{gathered}\text{base case true}\\\text{if P(k) -> P(k+1), then P(n) true for n >= base}\end{gathered}\]	\(P\): principal, perimeter, probability, or point P as used in the row \(k\): constant, scale factor, index, or number of successes \(n\): number of terms, sides, observations, trials, or periods Rule: Read the text condition before applying the symbols or calculator workflow.	Prove statements over integers.

26. Advanced K-12 / AP & IB Extension (27 formulae)

#	Formula name	Formula or rule	Variables / notation	Use
414	Derivative definition formula	\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]	\(f\): function f, frequency, or matrix entry \(h\): height, step size, or half-life interval \(x\): unknown, input variable, or x-coordinate	Define instantaneous rate of change.
415	Power rule	\[\frac{d}{dx}x^n=nx^{n-1}\]	\(d\): distance, diameter, divisor, or common difference \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Differentiate powers of x.
416	Product rule	\[(fg)'=f'g+fg'\]	\(f\): function f, frequency, or matrix entry \(g\): function g, greatest common factor, or gravitational/constant term	Differentiate a product.
417	Quotient rule	\[\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}\]	\(f\): function f, frequency, or matrix entry \(g\): function g, greatest common factor, or gravitational/constant term	Differentiate a quotient.
418	Chain rule	\[\frac{d}{dx}f(g(x))=f'(g(x))g'(x)\]	\(d\): distance, diameter, divisor, or common difference \(f\): function f, frequency, or matrix entry \(g\): function g, greatest common factor, or gravitational/constant term \(x\): unknown, input variable, or x-coordinate	Differentiate a composition.
419	Derivatives of trigonometric functions	\[d(\sin x)=\cos x; d(\cos x)=-\sin x; d(\tan x)=\sec^{2} x\]	\(d\): distance, diameter, divisor, or common difference \(x\): unknown, input variable, or x-coordinate	Differentiate basic trig functions.
420	Derivatives of exponential functions	\[d(e^{x})=e^{x}; d(a^{x})=a^{x} \ln a\]	\(a\): a base, initial value, or constant \(d\): distance, diameter, divisor, or common difference \(e\): Euler number for natural exponential growth, or a matrix entry \(x\): unknown, input variable, or x-coordinate	Differentiate exponential functions.
421	Derivatives of logarithmic functions	\[d(\ln x)=1/x; d(log_a x)=1/(x \ln a)\]	\(a\): a base, initial value, or constant \(d\): distance, diameter, divisor, or common difference \(x\): unknown, input variable, or x-coordinate	Differentiate logarithmic functions.
422	Implicit differentiation formula	\[dy/dx = -F_x/F_y for F(x,y)=0\]	\(F_x\): partial derivative with respect to the indicated variable \(F_y\): partial derivative with respect to the indicated variable \(F\): function, force, or named coefficient F \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Differentiate equations not solved for y.
423	Related rates formula	\[\begin{gathered}\text{differentiate the relation with respect to t}\end{gathered}\]	\(t\): time, parameter, or integration variable	Connect changing quantities over time.
424	Optimization formula	\[\begin{gathered}\text{critical points solve f'(x)=0 or f'(x) undefined}\end{gathered}\]	\(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Find maxima and minima under constraints.
425	Riemann sum formula	\[\sum f(x_i^*)\Delta x\]	\(x_i\): ith data value or sample value \(x_i\times \): sample point inside the ith subinterval \(\Delta x\): width of each subinterval or change in x \(f\): function f, frequency, or matrix entry \(i\): index value or matrix row index \(x\): unknown, input variable, or x-coordinate	Approximate area by rectangles.
426	Left Riemann sum formula	\[L_n = \sum from i=0 to n-1 f(a + i \Delta x)\Delta x\]	\(\Delta x\): width of each subinterval or change in x \(L\): length, left sum, or lower endpoint as used in the row \(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Approximate area using left endpoints.
427	Right Riemann sum formula	\[R_n = \sum from i=1 to n f(a + i \Delta x)\Delta x\]	\(\Delta x\): width of each subinterval or change in x \(R\): radius, right sum, range endpoint, or outer radius \(a\): a known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Approximate area using right endpoints.
428	Midpoint Riemann sum formula	\[M_n = \sum from i=1 to n f((x_(i-1)+x_i)/2)\Delta x\]	\(x_i\): ith data value or sample value \(\Delta x\): width of each subinterval or change in x \(M\): midpoint, matrix, mass, or midpoint sum \(f\): function f, frequency, or matrix entry \(i\): index value or matrix row index \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Approximate area using midpoints.
429	Trapezoidal rule	\[T_n = (\Delta x/2)(f(x0) + 2f(x1) + ... + 2f(x_(n-1)) + f(xn))\]	\(x_1\): x-1 coordinate of a point \(\Delta x\): width of each subinterval or change in x \(T\): time, trapezoidal estimate, or transformation T \(f\): function f, frequency, or matrix entry \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Approximate area with trapezoids.
430	Definite integral formula	\[\int_a^b f(x)\,dx=F(b)-F(a)\]	\(F\): function, force, or named coefficient F \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Compute signed area using an antiderivative.
431	Indefinite integral formula	\[\int f(x) dx = F(x) + C\]	\(C\): constant of integration \(F\): function, force, or named coefficient F \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Find a family of antiderivatives.
432	Fundamental theorem of calculus	\[\frac{d}{dx}\int_a^x f(t)\,dt=f(x),\quad \int_a^b f(x)\,dx=F(b)-F(a)\]	\(F\): antiderivative or function F \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(d\): distance, diameter, divisor, or common difference \(f\): function f, frequency, or matrix entry \(t\): time, parameter, or integration variable \(x\): unknown, input variable, or x-coordinate	Connect differentiation and integration.
433	Basic integration rules	\[\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1\]	\(C\): constant of integration \(n\): number of terms, sides, observations, trials, or periods \(x\): unknown, input variable, or x-coordinate	Integrate basic powers.
434	Integration by substitution	\[u = g(x), du = g'(x)dx\]	\(g\): function g, greatest common factor, or gravitational/constant term \(u\): substitution variable or vector u \(x\): unknown, input variable, or x-coordinate	Reverse the chain rule.
435	Area under a curve formula	\[A = \int_{a}^{b} f(x) dx, f(x) \ge 0\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(f\): function f, frequency, or matrix entry \(x\): unknown, input variable, or x-coordinate	Find area between a curve and the x-axis.
436	Area between curves formula	\[A = \int_{a}^{b} (top - bottom) dx\]	\(A\): area, amount, angle, event, or matrix named A as used in the row \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Find area between two curves.
437	Volume by disk method	\[V = \pi \int_{a}^{b} R(x)^{2} dx\]	\(\pi\): the constant pi, approximately 3.14159 \(R\): radius, right sum, range endpoint, or outer radius \(V\): volume, value, or vector V \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(x\): unknown, input variable, or x-coordinate	Find volume of revolution with no hole.
438	Volume by washer method	\[V = \pi \int_{a}^{b} (R(x)^{2} - r(x)^{2}) dx\]	\(\pi\): the constant pi, approximately 3.14159 \(R\): radius, right sum, range endpoint, or outer radius \(V\): volume, value, or vector V \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant \(r\): radius, rate, ratio, remainder, or root \(x\): unknown, input variable, or x-coordinate	Find volume of revolution with a hole.
439	Volume by shell method	\[V=2\pi\int_a^b \text{radius}\times\text{height}\,dx\]	\(\pi\): the constant pi, approximately 3.14159 \(V\): volume, value, or vector V \(a\): a known number, coefficient, side length, or constant \(b\): b known number, coefficient, side length, or constant	Find volume of revolution using cylindrical shells.
440	Differential equation separation formula	\[\frac{dy}{dx}=g(x)h(y)\Longrightarrow \int\frac{dy}{h(y)}=\int g(x)\,dx+C\]	\(C\): constant of integration \(g\): function g, greatest common factor, or gravitational/constant term \(h\): height, step size, or half-life interval \(x\): unknown, input variable, or x-coordinate \(y\): dependent variable, output, or y-coordinate	Solve separable first-order differential equations.

Frequently Asked Questions

This page includes 440 numbered formula entries across 26 K-12 and advanced high-school math topics, from number sense through AP and IB calculus readiness.

No. The early sections cover elementary and middle-school arithmetic, fractions, measurement, geometry, and probability. The later sections cover algebra, trigonometry, matrices, vectors, logic, and calculus readiness.

No. Use the page as a reference first. Prioritize formulas your class or exam requires, then practice substitution, units, graph meaning, and when the formula is valid.

Some math items are best stated as rules or structures, such as order of operations, congruent triangle rules, proof structures, and divisibility tests. They are included because students still use them like formula references.