Circle Theorems — GCSE Mathematics Revision

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What is Circle Theorems?

Circle theorems are rules about angles and lines in circles. The key theorems are: the angle at the centre is twice the angle at the circumference; angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles in a cyclic quadrilateral sum to 180°; a tangent meets a radius at 90°; and the alternate segment theorem. You must state the theorem you use to earn full marks.

Board notes: Circle theorems are Higher tier only on all boards. AQA and Edexcel frequently combine multiple theorems in a single question.

Worked example

Common mistakes

• 1Not stating which circle theorem you are using — examiners require the reason for each step.
• 2Confusing 'angle at the centre' with 'angle in a semicircle' — the semicircle case is when the angle at the centre is 180° (a diameter).
• 3Forgetting that the alternate segment theorem involves a tangent and a chord.
• 4Assuming two chords are equal without proof — only equal chords subtend equal angles.

Circle Theorems exam questions

Exam-style questions for Circle Theorems with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel and OCR specifications.

Frequently asked questions

• How many circle theorems do I need to know for GCSE?

  There are seven main circle theorems for GCSE Higher: angle at centre, angle in semicircle, angles in same segment, cyclic quadrilateral, tangent-radius, two tangents from a point, and alternate segment theorem.

• Do I need to prove circle theorems?

  You need to be able to apply them and state them as reasons. Formal proofs are not required at GCSE but understanding why they work helps with harder questions.