Slope Calculator

How to use the slope calculator

Choose Two points and enter the coordinates (x₁, y₁) and (x₂, y₂), or choose A point and the slope if you already know m. The calculator gives the slope as a reduced fraction and a decimal, tells you whether the line rises or falls, and works out the angle the line makes with the x-axis, the percent grade, the intercepts and the distance between the points. It also writes the equation of the line in slope-intercept, point-slope and standard form, with a small chart of the line.

The slope formula

Slope measures steepness: how much y changes for every one-unit increase in x. Once you have the slope, the rest follows:

Worked example

Find the line through (2, 3) and (6, 11).

1. Rise: 11 − 3 = 8. Run: 6 − 2 = 4.
2. Slope: m = 8 ÷ 4 = 2. For every step right, the line climbs two steps.
3. Intercept: b = 3 − 2 × 2 = −1.
4. Equation: y = 2x − 1. In point-slope form, y − 3 = 2(x − 2); in standard form, 2x − y = 1.
5. Angle: tan⁻¹(2) ≈ 63.43°; grade 200%.

The order of the points does not matter, as long as you subtract in the same order on the top and bottom: (3 − 11) ÷ (2 − 6) = −8 ÷ −4 = 2 as well.

Reading the slope

The larger the absolute value of m, the steeper the line. Two lines are parallel if they have the same slope, and perpendicular if their slopes multiply to −1 — the perpendicular slope to 2 is −½. The calculator lists the perpendicular slope in the equation table.

Forms of a linear equation

• Slope-intercept form, y = mx + b — best for graphing: start at b on the y-axis and move with slope m.
• Point-slope form, y − y₁ = m(x − x₁) — the quickest to write when you know one point and the slope.
• Standard form, Ax + By = C — uses whole-number coefficients, convenient for solving systems of equations and finding intercepts. The calculator reduces A, B and C to the smallest integers with A positive.

Slope in the real world

Slope describes the pitch of a roof (often given as rise per 12 units of run), the gradient of a road or railway, the steepness of a ramp or staircase, and any rate of change in a graph — speed on a distance–time graph, or cost per item on a price chart. In economics and science, the slope of a best-fit line is the key number linking two variables.

The straight-line distance between your two points is the hypotenuse of the rise-and-run triangle, which the distance calculator computes in 2D or 3D. To find the point halfway along the segment, use the midpoint calculator, and for the angles and sides of the right triangle formed by rise and run, try the Pythagorean theorem calculator. For staircases specifically, the stair calculator turns rise and run into step dimensions.