Work with circle equations in the coordinate plane (center-radius form). | Circles | SAT Math

Center-radius form shows the center and radius at a glance. Flip the signs inside the parentheses, and remember the right side is radius squared.

Core Idea

The standard equation of a circle is $(𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2 = 𝑟 2$ , where $(ℎ, 𝑘)$ is the center and $𝑟$ is the radius. Read the center and radius directly from this form.

Understanding

Rule: Center-radius form tells you everything at a glance.

$(𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2 = 𝑟 2$ means center $(ℎ, 𝑘)$ .
The signs inside the parentheses flip when you read the center.
The right side is $𝑟 2$ , so take a square root only when you need $𝑟$ .

Use that form both to read an equation and to build one from a center and point.

Step by Step

If needed, rearrange the equation into $(𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2 = 𝑟 2$ .
Read the center: $(ℎ, 𝑘)$ — flip the signs inside the parentheses.
Read the radius: $𝑟 = \sqrt r i g h t s i d e$ .
To write the equation: plug in the center and compute $𝑟 2$ from a known point.

Misconceptions

Reading $(𝑥 - 3) 2$ as center x-coordinate −3 instead of +3.
Reporting $𝑟 2$ as the radius instead of taking the square root.
Forgetting that $(𝑥 + 2) 2$ means the center's x-coordinate is −2.

Question

Worked Example

A circle has center $(3, - 4)$ and passes through the point $(6, 0)$ . What is the equation of the circle?

Select an answer to see the explanation