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Circle theorems - Higher - AQAAngles in a semicircle - Higher

Circles have different angle properties described by different circle theorems. Circle theorems are used in geometric proofs and to calculate angles.

Part ofMathsGeometry and measure

Angles in a semicircle - Higher

Click to explore updated revision resources for GCSE Maths: Angles in a semicircle, with step-by-step slideshows, quizzes, practice exam questions, and more!

The angle at the circumference in a is a right angle.

Triangle (ABP) within circle, passing through centre, O

Angle APB = 90°

Example

Calculate the angle \(z\).

Triangle (STU) within circle, passing through centre, O

The angle at the circumference in a semi-circle is 90°.

Angle STU = 90°

Angles in a triangle add up to 180°.

\(z = 180^\circ - 90^\circ - 31^\circ = 59^\circ\)

Proof

The angle on a straight line is 180°. The angle VOY = 180°.

Triangle (VWY) within circle, passing through centre, O

The angle at the centre is double the angle at the circumference.

Angle VWY = \(\frac{1}{2}\) of angle VOY = \(\frac{1}{2} \times 180\).

Angle VWY = 90°

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