# Circle Theorem: The Angle at the Center of the Circle Is Twice that at the Circumference

The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference.

# How to Use the Circle Theorem that the Angle at the Center Is Twice the Angle at the Circumference

Question: What is the angle θ in the circle below?

Step 1: Find the angle at the circumference.
The angle at the circumference is 40°.

Step 2: Multiply the angle at the circumference by 2.
2 × 40° = 80°.

The angle at the center of the circle is 80°.

# A Real Example of How to Use the Circle Theorem that the Angle at the Center Is Twice the Angle at the Circumference

In the previous example, the angle at the circumference is given so the angle at the center can be found.

In this example, the angle at the center is given so the angle at the circumference can be found.

Question: What is the angle θ in the circle below?

Step 1: Find the angle at the center.
The angle at the circumference is 100°.

Step 2: Divide the angle at the center by 2.
100° ÷ 2 = 50°.

The angle at the circumference of the circle is 50°.

# Another Real Example of How to Use the Circle Theorem that the Angle at the Center Is Twice the Angle at the Circumference

The slider below shows a real example of the circle theorem that the angle at the center of a circle is twice the angle at the circumference:
Geometry Lessons
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# USEFUL DEFINITIONS

An arc is a portion of the circumference.

The angle subtended by an arc is the angle made by lines joining the ends of an arc to a point.

The angle subtended by an arc at the center of the circle is the angle made by lines joining the ends of the arc to the center of the circle.

The angle subtended by an arc at the circumference of the circle is the angle made by lines joining the ends of the arc to any point on the circumference of the circle.