Slope calculator

Everything about the straight line through two points.

Formula Geometry

Slope (m)

Slope as a fraction: 3/4
Equation: y = 0.75x + 1.25
Angle: ≈ 36.8699 °
Gradient: 75%
Distance: 5
Midpoint: (3, 3.5)
y-intercept (b): 1.25

Step by step

m = (5 − 2) ÷ (5 − 1) = 3 ÷ 4 = 3/4
b = 2 − 0.75 × 1 = 1.25

How it works

The slope is the rise (change in y) divided by the run (change in x). A positive slope goes up from left to right; a negative one goes down.

The y-intercept b is where the line crosses the y-axis, found by putting one point into y = mx + b. The angle is the arctangent of the slope, and the gradient percentage is the slope times 100, as on road signs.

Calculations are exact, so slopes like 1/3 stay fractions.

Formula

m = (y₂ − y₁) ÷ (x₂ − x₁)
b = y₁ − m × x₁
angle = arctan(m)
d = √((x₂ − x₁)² + (y₂ − y₁)²)

Worked example

From (1, 2) to (5, 5): m = 3 ÷ 4 = 0.75, b = 2 − 0.75 × 1 = 1.25, so y = 0.75x + 1.25. The angle is about 36.87°, the distance 5 and the midpoint (3, 3.5).

Questions people ask

What is the slope of a horizontal line?

Zero: y does not change as x changes. Its equation is simply y = b.

Why is a vertical line’s slope undefined?

The run is zero, and division by zero is undefined. Its equation is x = a constant.

How is a 10% road gradient related to slope?

A 10% gradient is a slope of 0.1: 10 m of climb for every 100 m measured horizontally, an angle of about 5.71°.