This key facts covers The Eight Circle Theorems within Circle Theorems for GCSE Mathematics. Revise Circle Theorems in Geometry & Measures for GCSE Mathematics with 18 exam-style questions and 5 flashcards. This topic shows up very often in GCSE exams, so students should be able to explain it clearly, not just recognise the term. It is section 2 of 6 in this topic. Use this key facts to connect the idea to the wider topic before moving on to questions and flashcards.
The Eight Circle Theorems
| # | Theorem | Key Relationship |
|---|---|---|
| 1 | Centre vs Circumference | Angle at centre = 2× angle at circumference |
| 2 | Angle in Semicircle | Angle in semicircle = 90° (always!) |
| 3 | Same Segment | Angles in same segment are EQUAL |
| 4 | Cyclic Quadrilateral | Opposite angles add to 180° |
| 5 | Tangent ⊥ Radius | Tangent meets radius at 90° |
| 6 | Two Tangents | Tangents from external point are EQUAL |
| 7 | Alternate Segment | Angle between tangent & chord = angle in alternate segment |
| 8 | Perpendicular Bisector | Perpendicular from centre BISECTS the chord |
Practice questions for Circle Theorems
AB is a diameter of a circle. Point C lies on the circle. What is the size of angle ACB?
State the circle theorem used to find angle x in the diagram, and explain why it applies in this situation.
Quick recall flashcards
Cyclic Quadrilateral
If all 4 vertices lie on a circle, opposite angles sum to 180°. Also works in reverse: if opposite angles sum to 180°, then it's cyclic!
Angle in Semicircle
Any angle inscribed in a semicircle (with diameter as base) is always 90°. Always!