Angle in a Semicircle

What are circle theorems?

You will have learned a lot of angle facts for your GCSE, including angles in polygons and angles with parallel lines
Circle Theorems deal with angle facts that occur when lines are drawn within and connected to a circle

What do I need to know?

You must be familiar with the names of parts of a circle including radius, diameter, arc, sector, chord, segment and tangent

Parts of a circle, IGCSE & GCSE Maths revision notes

To solve some problems you may need to use the angle facts you are already familiar with from triangles, polygons, and parallel lines
You may also have to use the formulae for circumference and area, so ensure you’re familiar with them
- $Circumference = π \times diameter$ (C = πd)
- $Area = {πr}^{2}$ (A = πr²)

Circle theorem: The angle in a semicircle is a right angle

This circle theorem may be seen either in a full circle with a diameter drawn, or in a semicircle
It states that if a triangle is formed with two vertices at either end of a diameter and the third vertex on the circumference of the circle, then the third vertex will be a right angle (90°)
Look for a diameter in the circle and see if it makes the base of a triangle, with its top vertex at the circumference
- Make sure that you are looking at a diameter by checking it goes through the centre
- These questions only need half of the circle so they could appear in whole circles or in semicircles only
Any angle at the circumference that comes from each end of the diameter in this way will be 90°
This is most commonly known as the angle in a semicircle theorem, however if using it in an exam you must use the keywords
- The angle in a semicircle is 90°
Look out for triangles hidden among other lines/shapes within the circle

Examiner Tip

As soon as you spot this arrangement in a question, mark the angle as 90 degrees on the diagram
- Sometimes just doing this will earn you a mark!

Worked example

A, B, and C are points on a circle. AC is a diameter of the circle. Find the value of a.

angle-in-semicircle-diagram

As the line AC is the diameter of the circle, we can use the circle theorem "the angle in a semicircle is a right angle" to state that angle B must be 90 degrees

We can now use the fact that the internal angles of a triangle sum to 180, to find the unknown angle

a + 53 + 90 = 180

a + 143 = 180

a = 37

Angle in a Semicircle (CIE IGCSE Maths: Core)

Revision Note

What are circle theorems?

What do I need to know?

Angle in a Semicircle

Circle theorem: The angle in a semicircle is a right angle

Examiner Tip

Worked example

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Author: Amber