The Centre Angle Is Double The Circumference Angle (circle theorem math help).

If you make an angle at the centre of a circle then this angle will be double the circumference angle providing that the angles are made from the same two points on the circumference of the circle.

In your exam you can quote “the centre angle is double the circumference angle” if the exam question requires you to give reasons for your answer.

You can see this circle theorem in the picture to the right. The circumference angle is marked as x and the centre angle is marked as 2x. In some exam questions they might want you to find the other angle at the centre of the circle. To do this, subtract the first centre angle you have found from 360 degrees. You can always check that you have calculated the correct centre angle by seeing if the angle you have found is acute, obtuse or reflex as it should tie in with the diagram on the exam paper.

Let’s take a look at some examples.

Example 1

Work out angles x and y in the diagram below. Give reasons for your answer.

To calculate angle x double the circumference angle:

56 + 56 = 112 degrees. You can see this angle fits in with the diagram, as angle x is an obtuse angle (more than a right angle).

Angle y is the other angle at the centre of the circle. To get angle y subtract angle x from 360 degrees:

360 – 112 = 248 degrees.

All you need to do next is to write down the two angle facts that have been used:

The centre angle is double the circumference angle and angles around a point add up to 360 degrees.

Example 2

Work out angle x in the diagram shown below.

In this question you need to be careful which angle you half.

If you half the centre angle shown you get 122 degrees, but angle x cannot be 122 degrees as angle x is clearly acute from the diagram.

First you need to work out the other centre angle by subtracting 244 from 360:

360 – 244 = 116 degrees.

Now if you half this centre angle you get 58 degrees which looks a much better fit than than 122 degrees!

So angle x = 58 degrees.