Slope Calculator

Rise over run

Slope is the change in y divided by the change in x: (y₂ − y₁) / (x₂ − x₁). Positive means the line rises left to right, negative means it falls, zero means horizontal. A vertical line has an undefined slope rather than an infinite one, because the denominator is zero — which is why vertical lines cannot be written in y = mx + b form and need x = k instead.

Three ways to express the same steepness

A gradient of 0.5 is a ratio. As a percentage it is 50 percent, which is gradient times 100. As an angle it is arctan(0.5) ≈ 26.6 degrees. These are not interchangeable, and the confusion is common: a 100 percent grade is 45 degrees, not vertical. Road signs use percentage, roofing uses rise-per-12-inches, and mathematics uses the ratio.

Getting the line equation

With a slope and one point, point-slope form gives y − y₁ = m(x − x₁), which rearranges into y = mx + b. The intercept b is where the line crosses the y-axis, found by substituting either point back in. Two points are enough to define a line completely, which is why the slope calculation and the equation always come together.

Parallel and perpendicular

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is −1, so a line of slope 2 is perpendicular to one of slope −1/2. The negative reciprocal rule fails only for horizontal and vertical pairs, where one slope is zero and the other undefined — they are perpendicular, but the product rule cannot express it.

Practical gradients

Accessible ramps are commonly limited to 1:12, about 8.3 percent. Drainage typically wants a minimum fall of 1 to 2 percent so water moves without standing. Wheelchair-accessible paths, road maximum grades and roof pitches all use different conventions for the same underlying number, so checking which convention a specification uses matters more than the arithmetic.

Slope is a rate of change anywhere

Outside geometry, the slope of a line through two data points is the average rate of change between them — miles per hour, cost per unit, growth per year. That is also the foundation of the derivative, which is the slope as the two points converge. A regression line's slope is the same idea fitted across many points rather than two.