Coordinate geometry formulas

Coordinate geometry turns shapes into numbers: a point is a pair (x, y), a line is an equation, and lengths and angles come from arithmetic on coordinates. These are the formulas for working with two points, writing the equation of a line or a circle, and measuring how far a point is from a line.

Printable cheat sheet Formula finder

Distance formula

d = √((x₂ − x₁)² + (y₂ − y₁)²)

The straight-line distance between two points $(x₁, y₁)$ and $(x₂, y₂)$ : Pythagoras on a grid.

Midpoint formula

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The point exactly halfway between two points: average the x-coordinates and average the y-coordinates.

Slope formula

m = (y₂ − y₁)/(x₂ − x₁)

The steepness of the line through two points: rise over run.

Slope-intercept form

y = mx + b

The equation of a straight line from its slope m and the point b where it crosses the y-axis.

Point-slope form

y − y₁ = m(x − x₁)

The equation of a line through a known point with a known slope, without first finding the intercept.

Equation of a circle

(x − h)² + (y − k)² = r²

Every point (x, y) on a circle with centre (h, k) and radius r satisfies this equation.

Slopes of perpendicular lines

m₁·m₂ = −1 ⇔ m₂ = −1/(m₁)

Two lines are perpendicular when their slopes multiply to −1: flip the slope and change its sign.

Distance from a point to a line

d = (|ax₀ + by₀ + c|)/(√(a² + b²))

The shortest distance from a point $(x₀, y₀)$ to the line $ax + by + c = 0$ , measured along the perpendicular.