Triangle Calculator

Name: Triangle Calculator
Author: Jitendra Kumar

Solve triangle sides, angles, area, perimeter, height, medians, bisectors, inradius, circumradius, centers, slopes, and step-by-step formulas from multiple input modes.

Last Updated: May 16, 2026

Length unit

Area unit

Side a

Side b

Side c

Show diagram grid

Live triangle diagram

Sides follow the standard convention: side a is opposite angle A.

Solved triangle

Scalene Acute

Area

14.6969 cm2

Perimeter

18 cm

Semiperimeter

9 cm

Triangle type

Scalene Acute

Side a

5 cm

Side b

6 cm

Side c

7 cm

Validity check

Valid

Sides and angles

Element	Length	Degrees	Radians
Side a (BC)	5 cm	44.4153 deg	0.775193 rad
Side b (CA)	6 cm	57.1217 deg	0.996961 rad
Side c (AB)	7 cm	78.463 deg	1.369438 rad

Altitudes, medians, bisectors

Metric	a	b	c
Altitudes	h_a 5.8788 cm	h_b 4.899 cm	h_c 4.1991 cm
Medians	m_a 6.0208 cm	m_b 5.2915 cm	m_c 4.272 cm
Angle bisectors	t_a 5.9822 cm	t_b 5.1235 cm	t_c 4.2251 cm

Centers

Center	Coordinate	Meaning
Centroid	(3.7619, 1.3997) cm	Average of the three vertices
Incenter	(4, 1.633) cm	Center of the incircle
Circumcenter	(3.5, 0.7144) cm	Center of the circumcircle
Orthocenter	(4.2857, 2.7703) cm	Intersection of altitudes

Radius and slope checks

Circle metric	Value	Formula context
Inradius	1.633 cm	Incircle radius
Circumradius	3.5722 cm	Circumcircle radius
Incircle area	8.3776 cm2	pi r^2
Circumcircle area	40.088 cm2	pi R^2

Side	Slope
AB	0
BC	-1.547046
CA	0.979796

Step-by-step solution

1. Check triangle inequality
Formula: \(a + b > c, a + c > b, b + c > a\)
Substitution: \(5 + 6 > 7, 5 + 7 > 6, 6 + 7 > 5\)
Result: All three checks must be true before solving.
The calculator rejects side lengths that cannot close into a triangle.
2. Find the semiperimeter
Formula: \(s = (a + b + c) / 2\)
Substitution: \(s = (5 + 6 + 7) / 2\)
Result: s = 9
Heron formula uses half the perimeter as a compact area input.
3. Apply Heron formula
Formula: \(Area = \sqrt{s(s-a)(s-b)(s-c)}\)
Substitution: \(Area = \sqrt{9(9 - 5)(9 - 6)(9 - 7)}\)
Result: Area is calculated from the three sides.
This solves area without needing a separate height measurement.

Geometry Learning Notice

This calculator is for education, homework checking, planning, and geometric estimation. It assumes standard Euclidean geometry. For surveying, structural engineering, regulated design, or safety-critical layout work, verify dimensions with project-specific standards and instruments.

Reviewed For Methodology, Labels, And Sources

Every CalculatorWallah calculator is published with visible update labeling, linked source references, and review of formula clarity on trust-sensitive topics. Use results as planning support, then verify institution-, policy-, or jurisdiction-specific rules where they apply.

Reviewed by Jitendra Kumar, Founder & Editorial Standards Lead. Page updated May 16, 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

How to Use This Calculator

Start by matching the calculator mode to the information you already know. Three sides belong in SSS or Heron mode. Two sides and the angle between them belong in SAS mode. Two angles and one side belong in ASA/AAS mode. Coordinate points belong in the coordinate mode, where the diagram points can also be dragged directly.

The side labels follow the standard convention: side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\). Keeping that convention consistent is what lets the Law of Sines, Law of Cosines, medians, bisectors, and radius formulas line up correctly.

Use the unit selectors before reading the final answer. Length outputs use the selected length unit, while area outputs convert independently to square units, acres, or hectares. That makes the same solved triangle useful for geometry homework, land area, construction, and design planning.

Step 1: Choose the input mode
Pick SSS, SAS, ASA/AAS, right triangle, coordinates, base-height, or another mode that matches the values in your problem.
Step 2: Enter the known values
Type side lengths, angles, coordinates, radius, area, median, or altitude values using the selected unit system.
Step 3: Check the validity message
The solver verifies triangle inequality, angle limits, positive dimensions, and collinear-coordinate errors before returning results.
Step 4: Read the output tables
Review area, perimeter, side lengths, angles, altitudes, medians, bisectors, centers, slopes, inradius, and circumradius.
Step 5: Follow the step-by-step solution
Use the formula, substitution, simplification, and final answer chain to understand how the result was calculated.
Step 6: Copy or print the result
Copy the result summary for notes, or use Print / Save PDF when you need a clean solution record.

How This Calculator Works

The calculator first converts the selected input mode into a complete triangle when enough independent information is available. SSS and Heron modes start from the three sides. SAS mode uses the Law of Cosines. ASA/AAS mode uses the angle-sum rule and Law of Sines. Coordinate mode uses the distance formula and shoelace area formula.

Before solving, the engine checks triangle validity. Side-based modes must satisfy \(a+b>c\), \(a+c>b\), and \(b+c>a\). Angle-based modes must keep the angle sum at \(180^\circ\). Coordinate mode rejects collinear points because a zero-area triangle has no meaningful inradius, circumcenter, or orthocenter.

Once the triangle is valid, the solver calculates area, perimeter, semiperimeter, angles in degrees and radians, altitudes, medians, angle bisectors, inradius, circumradius, incircle and circumcircle areas, centroid, orthocenter, circumcenter, incenter, side slopes, and triangle type. The step panel shows the formula and substitution that produced the result for the selected mode.

Triangle Formulas, Examples, and Common Mistakes

Formula library

Formula	Expression	When to use it
Area from base and height	\(A = \frac{1}{2}bh\)	Use when a perpendicular height is known.
Semiperimeter	\(s = \frac{a+b+c}{2}\)	Half the perimeter; needed for Heron formula.
Heron formula	\(A = \sqrt{s(s-a)(s-b)(s-c)}\)	Find area from three side lengths.
Perimeter	\(P = a+b+c\)	Add the three side lengths.
Pythagorean theorem	\(c^2 = a^2+b^2\)	Use for right triangles.
Law of Sines	\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)	Use with ASA/AAS style information.
Law of Cosines	\(c^2=a^2+b^2-2ab\cos C\)	Use with SSS or SAS information.
Inradius	\(r = \frac{A}{s}\)	Radius of the incircle.
Circumradius	\(R = \frac{abc}{4A}\)	Radius of the circle through all vertices.

Solved examples

Example	Setup	Solution
Base-height area	Given \(b=10\) and \(h=6\), use \(A=\frac{1}{2}bh\).	\(A=\frac{1}{2}\times 10\times 6=30\) square units.
Heron formula	Given \(a=5\), \(b=6\), \(c=7\), first find \(s=9\).	\(A=\sqrt{9(9-5)(9-6)(9-7)}=\sqrt{216}\approx 14.7\).
SAS missing side	Given \(b=8\), \(c=10\), and \(A=45^\circ\), use Law of Cosines.	The calculator solves side a first, then computes the remaining angles and centers.

Triangle type identification

Type	Side rule	Angle rule
Equilateral	All three sides are equal.	All angles are 60 degrees.
Isosceles	Two sides are equal.	The base angles are equal.
Scalene	All three sides are different.	Angles are usually all different.
Acute	Largest angle is less than 90 degrees.	All angles are acute.
Right	One angle is exactly 90 degrees.	Pythagorean theorem applies directly.
Obtuse	One angle is greater than 90 degrees.	The longest side sits opposite the obtuse angle.

Real-life uses

Use case	How triangle solving helps
Land and plot area	Estimate triangular land sections, garden beds, and irregular lot pieces.
Construction layout	Check diagonals, roof framing, brace lengths, and layout triangles.
Architecture and design	Model triangular faces, supports, slopes, and visual proportions.
Navigation and physics	Resolve vector components, coordinate distances, and triangular paths.
Geometry homework	Verify side, angle, area, inradius, circumradius, and center calculations.
Trigonometry practice	Compare Law of Sines, Law of Cosines, Heron formula, and right-triangle formulas.

Common mistakes

Mistake	How to avoid it
Using the wrong mode	SAS needs the included angle. If the angle is not between the known sides, use a sine-law setup instead.
Skipping triangle inequality	Three side lengths must pass a + b > c, a + c > b, and b + c > a.
Mixing degrees and radians	Most classroom triangle problems use degrees. The calculator shows both degrees and radians in the output.
Assuming base and height determine every triangle	Base and height determine area, but the apex offset determines the final side lengths.
Rounding too early	Keep more digits until the final answer, especially for trigonometric and coordinate problems.

If your problem is specifically a right triangle, the Pythagorean Theorem Calculator gives a more focused right-triangle workflow. If the problem starts from coordinate points and line steepness, the Slope Calculator is a useful companion.

Keep the research moving with Pythagorean Theorem Calculator, Slope Calculator, Scientific Calculator, and Area Converter.

Frequently Asked Questions

It solves triangles from SSS, SAS, ASA/AAS, right-triangle inputs, base and height, coordinates, area plus side, side plus median, equilateral, isosceles, and radius plus angle inputs.

For side lengths, it checks that a + b > c, a + c > b, and b + c > a. For angle-based modes, it checks that the angles are positive and add to less than or equal to 180 degrees with the third angle computed from the remainder.

Use Heron formula. First find the semiperimeter s = (a + b + c) / 2, then calculate area with sqrt(s(s-a)(s-b)(s-c)).

Use the Law of Cosines when you know two sides and the included angle, or when you need angles from three known sides.

Use the Law of Sines when you know one side-angle pair and another angle, as in ASA or AAS triangle problems.

Yes. Enter the x and y coordinates for points A, B, and C, or drag the points on the diagram. The calculator finds side lengths, slopes, area, angles, centers, and radii.

The inradius is the radius of the circle tangent to all three sides inside the triangle. The circumradius is the radius of the circle that passes through all three vertices.

Yes. The solution panel lists the given values, formula, substitution, simplification, and final result for the selected solving mode.

Yes. Length outputs support mm, cm, m, km, inches, feet, yards, and miles. Area outputs support square metric units, square imperial units, acres, and hectares.

Yes. The calculator is free to use for homework checking, geometry practice, layout planning, construction estimates, and coordinate-geometry verification.