Circle Theorems: The Eight Rules That Cover Every Question
May 9, 2026 · 5 min · circle theorems · geometry · GCSE math
Circle theorems trip up GCSE and IGCSE students because they sound abstract on paper. Drawn on a circle, they're obvious. Spend an hour with a compass and a ruler and you've got them.
The eight you need
- Angle at the centre = 2 × angle at the circumference (when both angles stand on the same arc)
- Angle in a semicircle is 90° (special case of rule 1, where the centre angle is 180°)
- Angles in the same segment are equal
- Cyclic quadrilateral: opposite angles sum to 180°
- Tangent meets radius at 90°
- Two tangents from a point are equal length
- Alternate segment theorem — angle between tangent and chord equals angle in the alternate segment
- Perpendicular from centre bisects chord
How to learn them
- Draw each one once, by hand, with a compass
- Label every angle and side
- Photograph it for revision later
That fifteen minutes does more than reading the textbook five times.
How to spot which to use
The exam question gives you a circle with some angles labelled. Look at:
- Where the angles sit (centre vs circumference vs same arc)
- Whether you see a tangent
- Whether you see a cyclic quadrilateral (four points on the circle)
Those three clues will point you at the right theorem in seconds.
Common pitfalls
- Mixing up theorems 1 and 3 (centre vs same segment)
- Forgetting tangents must touch the circle without crossing it
- Missing that cyclic quadrilateral angles only sum to 180° if all four corners touch the circle