Slope Calculator

x₁

y₁

x₂

y₂

Result

Gradient2
Equation of the liney = 2x + 0
Intercept0
Angle to x-axis, °63.4349
Grade, %200
Perpendicular gradient-0.5

Rise over run

The gradient, the line's equation in y = mx + b form, its angle with the x-axis, the grade as a percentage and the perpendicular gradient.

The gradient says how far a line climbs for each step to the right. It is the change in y divided by the change in x, and from it the line's equation falls out immediately.

The same number is what civil engineers call a grade, expressed as a percentage. A 5 percent grade means climbing five metres over a hundred travelled — a gradient of 0.05.

What this calculator shows

The gradient and the completed equation of the line
The angle to the x-axis and the grade as a percentage
The gradient of any line perpendicular to it

What to keep in mind

A vertical line is handled separately: its gradient is undefined and its equation is x = constant.
The perpendicular gradient is the negative reciprocal — the product of perpendicular gradients is always −1.

FAQs

What does a negative gradient mean?

The line falls from left to right. The larger the magnitude, the steeper the descent. Zero means a horizontal line.

Why is a vertical line's gradient undefined?

Because the change in x is zero and division by zero is undefined. Such a line is written x = constant and stands at 90 degrees.

How do gradient and angle relate?

The gradient is the tangent of the angle. A gradient of 1 is 45 degrees; a gradient of 2 is about 63.4.

How do I find a perpendicular line?

Take the negative reciprocal. For m = 2 the perpendicular has m = −0.5, and the product is exactly −1.

Worked example

Through (1, 2) and (4, 8)

Input: x₁=1, y₁=2, x₂=4, y₂=8

Output: m = 2, y = 2x + 0, angle 63.43°

Note: A 200 percent grade would be absurd for a road: wheelchair ramps are limited to about 8 percent, a gradient of 0.08.

Useful links

All tools in Math & Statistics

Slope Calculator

Result

Related tools

Distance Between Points

Linear Equation Solver

Roofing Calculator

Useful links