IGCSE Maths / Circle theorems

The angle in a semicircle revision notes

Edexcel International GCSE Maths (4MA1), chapter 17: Circle theorems, specification point 4.6c. Free lesson.

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A diameter subtends a right angle at the circumference. Spot the diameter, use the angle in a semicircle rule, then use the triangle angle sum if you need another angle.

Why a diameter gives a right angle

A diameter passes through the centre and joins two points on the circle. The angle at the centre is twice the angle at the circumference when both stand on the same arc. If the angle at the circumference is 90°, the angle at the centre is 180°, so the two ends of the line lie on a straight line. This gives the rule in reverse: an angle standing on a diameter is 90°. An arc is a piece of the circle's edge. A chord joins two points on the circle. It only gives this right angle if it passes through the centre, so it is a diameter.

How to solve a semicircle angle question

First check that one side of the triangle really passes through the centre. Then mark the right angle at the triangle's third corner. If the question asks for a different angle, use the fact that the angles in a triangle add to 180°. Write a reason for each step. The angle in a semicircle gives the reason for 90°; the triangle angle sum gives the reason for the remaining angle.

Check which angle the question asks for

The right angle is at the point on the circle opposite the diameter. It is not automatically the answer. Read the angle labels carefully: if the unknown is at another corner, finish the calculation using the triangle angle sum. Three corners on a circle do not automatically make a triangle with a right angle. One side must be a diameter. A line that misses the centre is only a chord and does not establish a 90° angle.

Use the same method with different labels

For a triangle formed by diameter ST and point R on the circle, the angle at R is 90°. If ∠ RST is 27°, subtract both known angles from 180° to find ∠ RTS.

More lessons in Circle theorems

• Parts of a circle and cyclic quadrilaterals
• Tangent and radius, and two tangents from a point
• The angle at the centre is twice the angle at the circumference
• Angles in the same segment
• Opposite angles of a cyclic quadrilateral
• The alternate segment theorem
• Using the alternate segment theorem in a reasoning chain
• Reasoning chains with circle theorems
• Multi-step circle theorem problems
• Intersecting chords inside a circle
• Intersecting chords from a point outside the circle