Circle Theorems

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Corbett MathsCircle theorems — all theorems with examples

GCSE Maths TutorAll of the Circle Theorems in 10 Minutes!! | Circle Theorem Series Part 1 | GCSE Maths Tutor

Key facts & methods

The eight circle theorems

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

Giving reasons in proofs

You MUST state the theorem name, not just the angle. AQA requires reasons.
E.g. "angle AOB = 2 × angle ACB (angle at centre is twice angle at circumference)".
E.g. "angle in a semicircle = 90°" — not just "right angle".
E.g. "opposite angles in a cyclic quadrilateral sum to 180°".
A common trap: the centre theorem only applies when both angles are subtended by the SAME arc.

Exam questions — 5 questions · 12 marks · Edexcel 4MA1 style

1 markAngle at centre4MA1 style

Point O is the centre of a circle. Points A, B, C are on the circumference. Angle ACB = 38°. What is angle AOB?

A 38°

B 76°

C 52°

D 142°

1 markCyclic quadrilateral4MA1 style

ABCD is a cyclic quadrilateral. Angle A = 73°. Find angle C.

A 62°

B 107°

C 73°

D 253°

4 marksCircle theorem proofGrade 7 Booklet style ✏️ No calculator

In the diagram, O is the centre of the circle. Points A, B, C lie on the circle. Angle OAB = 25°. Find angle ACB. Give reasons for each step of your working. (4 marks)

Hint: OA=OB (radii) → isosceles triangle. Find angle AOB. Then use: angle at centre = 2 × angle at circumference. State each theorem by name.

+40 XP

1 markTangent perpendicular4MA1 style

TA is a tangent to a circle at point A. O is the centre. Angle OAT = ?

A 45°

B 90°

C 60°

D Cannot be determined without more information

1 markAlternate segment theorem4MA1 style

In the alternate segment theorem, the angle between a tangent to a circle and a chord drawn from the point of tangency equals:

A 90°

B The angle subtended by the chord at the centre

C The angle in the alternate segment

D Half the reflex angle at the centre

Circle Theorems

The eight circle theorems

Giving reasons in proofs

Module complete! 🎉