Circle theorems

Circle theorems are rules about angles and lines within circles. There are seven key theorems you must know for GCSE Higher, and you must state the theorem used in your working.

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Key facts to remember

1Angle at centre = 2 × angle at circumference (same arc).
2Angles in the same segment are equal.
3Angle in a semicircle = 90° (angle subtended by a diameter).
4Opposite angles in a cyclic quadrilateral sum to 180°.
5Tangent to a circle is perpendicular to the radius at the point of contact.
6Tangents from an external point are equal in length.
7Alternate segment theorem: angle between tangent and chord = angle in alternate segment.

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Worked examples

Example 1

O is the centre of a circle. Angle AOB = 112°. Find angle ACB, where C is a point on the major arc.

Working

Angle at centre = 2 × angle at circumference
Angle ACB = 112° ÷ 2 = 56°

Answer56°

Example 2

ABCD is a cyclic quadrilateral. Angle A = 78° and angle B = 105°. Find angles C and D.

Working

Opposite angles in a cyclic quadrilateral sum to 180°
Angle C = 180° − 78° = 102°
Angle D = 180° − 105° = 75°

AnswerC = 102°, D = 75°

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Common mistakes

✗Confusing angle at the centre with angle at the circumference — the centre angle is always double.

✗Not stating the circle theorem used — examiners require the reason.

✗Assuming all angles in the same circle are equal (they must be subtended by the same arc).

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Exam tips

✓Learn all seven theorems by name and always write the full theorem as your reason.

✓Mark the centre O clearly and identify which arc the angles are subtended by before applying theorems.

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