The Equation of a Circle

Every circle can be described by a single equation built from just two things: its center and its radius. The equation comes straight from the distance formula — every point on the circle is exactly $r$ away from the center.

Theorem Center and radius form

A circle with center $(h, k)$ and radius $r$ is the set of all points $(x, y)$ at distance $r$ from the center.

By the distance formula, that condition is:

(x-h)^2 + (y-k)^2 = r^2

Note Two things to watch

The sign of each center coordinate is reversed inside the equation. A center of $(1, -1)$ gives $(x-1)^2 + (y+1)^2$ — the minus becomes a plus.

The right-hand side is the radius squared, $r^2$ , not $r$ . A radius of 4 gives 16.

Example Writing and reading the equation

Center (3, 2), radius 4

Both coordinates are positive, so both signs flip to minus.

(x-3)^2 + (y-2)^2 = 4^2

⟹ (x − 3)² + (y − 2)² = 16.

Center (1, −1), radius 2

The −1 flips to +1, so the

y

term becomes

(y+1)

(x-1)^2 + (y+1)^2 = 2^2

⟹ (x − 1)² + (y + 1)² = 4.

Center at the origin, radius 5

Here

h = k = 0

, so both terms are just

x^2

and

y^2

⟹ x² + y² = 25.

Reading an equation backwards

Given

(x+2)^2 + (y-3)^2 = 9

, flip the signs back for the center.

Center

= (-2, 3)

, and

r = \sqrt{9}

⟹ center (−2, 3), radius 3.

Reference Forms and the sign rule

Case	Equation
Center (h, k)	(x − h)² + (y − k)² = r²
Center at origin (0, 0)	x² + y² = r²
h positive in the center	(x − h) in the equation
h negative in the center	(x + \|h\|) in the equation

Summary

General form: $(x-h)^2 + (y-k)^2 = r^2$ , where $(h, k)$ is the center and $r$ the radius.
At the origin it becomes $x^2 + y^2 = r^2$ , since $h = k = 0$ .
Sign rule: the center's sign is reversed in the equation — a negative coordinate becomes a plus.
The right side is always $r^2$ (the radius squared), never $r$ .