GCSE Maths: Circle Theorems Guide

Angles in a circle

The angle at the centre is twice the angle at the circumference standing on the same arc. The angle in a semicircle is always 90°.

Angles in the same segment

Angles subtended by the same arc, in the same segment, are equal to each other.

Cyclic quadrilaterals

Opposite angles in a cyclic quadrilateral (a quadrilateral with all four vertices on a circle) add up to 180°.

Tangents

A tangent meets a circle at exactly one point and is always perpendicular to the radius at that point. Two tangents drawn from the same external point are equal in length.

Common mistakes

• Applying a circle theorem to angles that don't actually meet its conditions (e.g. not on the same arc).
• Forgetting the tangent-radius angle is always exactly 90°.
• Not showing full reasoning — circle theorem answers usually need the theorem named as justification.