This topic summary covers Knowledge Organiser: 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 6 of 6 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Circle Theorems
Key Terms
- Cyclic quadrilateral: A quadrilateral with all 4 vertices on the circle
- Tangent: A line touching the circle at exactly one point
- Chord: A straight line joining two points on the circumference
- Alternate segment: The segment on the other side of a chord from the tangent
Must-Know Theorems
- Angle at centre = 2 × angle at circumference (same arc)
- Angle in a semicircle = 90°
- Angles in same segment are equal
- Opposite angles in a cyclic quadrilateral add to 180°
- Tangent meets radius at 90°
- Two tangents from external point are equal in length
- Alternate segment theorem: tangent-chord angle = angle in alternate segment
- Perpendicular from centre bisects a chord
Key Methods
- Always mark known angles and look for isosceles triangles (two radii = two equal sides)
- State the theorem used — required for full marks
- Two radii always form an isosceles triangle
Common Mistakes
- Not stating the theorem: Examiners require the theorem name as a reason — "angle in a semicircle = 90°" must be written, not just the answer
- Confusing angle at centre and angle at circumference: The angle at the centre is DOUBLE the angle at the circumference — students sometimes reverse this and halve the wrong angle
- Missing the isosceles triangle: Two radii form an isosceles triangle — mark the equal base angles to unlock further angle calculations
- Cyclic quadrilateral rule: Opposite angles in a cyclic quadrilateral sum to 180°, not 360° — add just the two opposite angles when setting up your equation
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!