How to Calculate Slope (& What the Number Means)

Slope measures how steep a line is — how much it rises or falls for each unit of horizontal movement. It is described as "rise over run": the vertical change divided by the horizontal change between any two points on the line.

Slope appears in algebra (describing linear equations), calculus (as the derivative), engineering (road grades, roof pitches), and everyday situations (wheelchair ramp inclines, hiking trail steepness).

The symbol for slope is m, which appears in the standard line equation y = mx + b. The origin of "m" for slope is debated among mathematicians — one common explanation is that it comes from the French word monter (to climb).

Given two points on a line, (x₁, y₁) and (x₂, y₂):

m = (y₂ − y₁) ÷ (x₂ − x₁)

This is also written as:

m = Δy ÷ Δx = rise ÷ run

Where Δ (delta) means "change in." The numerator is the vertical change (rise) and the denominator is the horizontal change (run).

Note: the order of the points does not affect the value of the slope (you get the same answer either way), but you must subtract the coordinates in the same order in both the numerator and denominator.

1. Identify two points on the line: (x₁, y₁) and (x₂, y₂).
2. Subtract the y-coordinates: rise = y₂ − y₁.
3. Subtract the x-coordinates: run = x₂ − x₁.
4. Divide rise by run: m = rise ÷ run.
5. Simplify the fraction if possible.

Check your answer: plot the two points and draw the line. A positive slope goes up to the right; a negative slope goes down to the right.

Example 1 — Positive slope

Points: (1, 2) and (4, 8)

• Rise = 8 − 2 = 6
• Run = 4 − 1 = 3
• Slope = 6 ÷ 3 = 2
• Interpretation: for every 1 unit right, the line rises 2 units.

Example 2 — Negative slope

Points: (0, 5) and (3, −1)

• Rise = −1 − 5 = −6
• Run = 3 − 0 = 3
• Slope = −6 ÷ 3 = −2
• Interpretation: for every 1 unit right, the line drops 2 units.

Example 3 — Zero slope

Points: (2, 4) and (7, 4)

• Rise = 4 − 4 = 0
• Run = 7 − 2 = 5
• Slope = 0 ÷ 5 = 0
• Interpretation: horizontal line, no rise.

Example 4 — Undefined slope

Points: (3, 1) and (3, 6)

• Rise = 6 − 1 = 5
• Run = 3 − 3 = 0
• Slope = 5 ÷ 0 = undefined (division by zero)
• Interpretation: vertical line.

• Positive slope (m > 0) — line rises from left to right. The larger the value, the steeper the rise.
• Negative slope (m < 0) — line falls from left to right. The more negative the value, the steeper the descent.
• Zero slope (m = 0) — perfectly horizontal line. y does not change as x changes.
• Undefined slope — perfectly vertical line. x does not change; you cannot divide by zero.
• Slope of 1 or −1 — 45° angle. Equal rise and run.
• Slope between 0 and 1 — line is more horizontal than diagonal (gentle incline).
• Slope greater than 1 — line is more vertical than diagonal (steep incline).

Once you know the slope of a line, you can write its full equation using slope-intercept form:

y = mx + b

Where m is the slope and b is the y-intercept (the y value where the line crosses the y-axis, i.e., where x = 0).

To find b when you know slope and one point (x₁, y₁):

b = y₁ − m × x₁

Example: slope m = 2, point (1, 5):

• b = 5 − (2 × 1) = 5 − 2 = 3
• Equation: y = 2x + 3

• Road grades: A 6% grade means a slope of 0.06 — the road rises 6 feet for every 100 feet of horizontal distance. Highway signs warn of steep grades for truck braking purposes.
• Roof pitch: A 4:12 pitch (common for residential roofs) means 4 inches of rise per 12 inches of run — a slope of 0.333.
• ADA ramp compliance: The Americans with Disabilities Act requires wheelchair ramps to have a maximum slope of 1:12 (1 inch rise per 12 inches run) — a slope of 0.083.
• Economics: The slope of a supply or demand curve shows the rate of change in quantity per unit change in price.
• Physics: The slope of a distance-time graph is velocity. The slope of a velocity-time graph is acceleration.

Use the slope calculator to find slope, angle, distance, midpoint, and the full line equation from any two points — without manual calculation.