Equation of a Circle

Center-radius form, reading off center and radius, completing the square.

The idea

The equation of a circle describes a circle in the coordinate plane by its center and radius. A circle with center $(h, k)$ and radius $r$ is the set of all points at distance $r$ from the center.

Theorem (Equation of a circle).

The circle with center $(h, k)$ and radius $r$ consists of exactly the points $(x, y)$ satisfying the center-radius form $(x-h)^{2} + (y-k)^{2} = r^{2}.$

By the distance formula, a point $(x, y)$ lies at distance $\sqrt{(x-h)^{2} + (y-k)^{2}}$ from $(h, k)$. Setting this distance equal to $r$ and squaring both sides to remove the root gives the form above.

The form subtracts the coordinates of the center, so a plus sign inside a bracket signals a negative coordinate: $x + 4$ means $x - (-4)$, and $h = -4$. The right side is the radius squared, so recovering $r$ takes a square root — the positive one, since a radius is a length.

When the brackets are multiplied out and like terms collected, the center and radius are hidden. Completing the square in $x$ and in $y$ rebuilds the brackets and restores the form.

Ways to work on it

• Walkthrough. Build the center-radius form, then read center and radius back out.
• Practice. Read the center and radius off a circle in center-radius form.
• Hardest. Find the center and radius of a circle given in expanded general form.

Not sure where to start? Take the ten-question placement test.