Use the distance formula to connect a point on the circle to its radius. | Circles | SAT Math

The distance formula and the circle equation express the same relationship. Use the center and one point to get the radius or the squared radius.

Core Idea

A point $(𝑥, 𝑦)$ is on a circle with center $(ℎ, 𝑘)$ and radius $𝑟$ exactly when $\sqrt (𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2 = 𝑟$ . The distance formula and the circle equation are the same relationship.

Understanding

Rule: The circle equation is just the distance formula written with $𝑟 2$ .

Use the center and a point to find the radius.
If the question asks for $𝑟 2$ , stop before taking the square root.
Compare the distance to $𝑟$ to decide whether a point is inside, on, or outside the circle.

The distance formula and the circle equation are the same relationship in different forms.

Step by Step

Identify the center $(ℎ, 𝑘)$ and the given point $(𝑥, 𝑦)$ .
Compute the distance: $𝑑 = \sqrt (𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2$ .
If $𝑑 = 𝑟$ , the point is on the circle. If $𝑑 < 𝑟$ , inside. If $𝑑 > 𝑟$ , outside.
To build the circle equation, use $𝑟 2 = (𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2$ .

Misconceptions

Subtracting coordinates in the wrong order — it doesn't matter for distance (squaring eliminates the sign), but watch out for errors in the intermediate steps.
Forgetting to square the differences — computing $| 𝑥 - ℎ | + | 𝑦 - 𝑘 |$ instead of $\sqrt (𝑥 - ℎ) 2 + (𝑦 - 𝑘) 2$ .
Confusing $𝑟$ with $𝑟 2$ when plugging into the circle equation.

Question

Worked Example

The center of a circle is $(1, 2)$ and the point $(4, 6)$ is on the circle. What is $𝑟 2$ ?

Select an answer to see the explanation