Use circle definitions and basic properties (radius, diameter, chords) to relate lengths. | Circles | SAT Math

Radius, diameter, and chord questions usually turn into a right triangle from the center to a chord. Diameter is always twice the radius.

Core Idea

The radius goes from center to edge, the diameter is twice the radius and passes through the center, and a chord connects two points on the circle. A diameter is the longest possible chord.

Understanding

Rule: Radius, diameter, and chord are different pieces of the same circle.

Radius: center to edge.
Diameter: $2 𝑟$ , and the longest chord.
A perpendicular from the center to a chord bisects that chord.

Draw the radius first; it often creates the right triangle you need.

Step by Step

Draw the radius to any relevant point on the circle.
If a chord is involved, draw a perpendicular from the center to the chord to create right triangles.
Use the Pythagorean theorem with the radius, half-chord, and perpendicular distance.
Remember: diameter = $2 𝑟$ .

Misconceptions

Confusing radius with diameter — diameter is $2 𝑟$ , not $𝑟$ .
Assuming a chord that looks like it passes through the center actually does — check the problem statement.
Forgetting that a perpendicular from the center to a chord bisects the chord.

Question

Worked Example

A chord is 10 units long and sits 12 units from the center of a circle. What is the radius of the circle?

Select an answer to see the explanation