The Lesson

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 where the Angle at the Center Is Twice the Angle at the Circumference

Question

What is the angle θ in the circle below?

Step-by-Step:

1

Find the angle at the circumference. In our example, angle at the circumference is 40°.

2

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

Answer:

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

Another Real Example of How to Use the Circle Theorem where 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-by-Step:

1

Find the angle at the center. In our example, angle at the center is 100°.

2

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

Answer:

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

Lesson Slides

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.

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.