Sheet № 52 · Higher only · AQA · Edexcel · OCR
Circle Theorems –
Circle theorems are one of the most recognisable Higher tier topics in GCSE Maths and appear on every exam board — AQA, Edexcel, and OCR. Questions typically ask you to find missing angles in diagrams involving circles, chords, tangents, and cyclic quadrilaterals. You need to know the theorems, spot which ones apply, and justify your answ
§Key definitions
Question:
Points A, B, and C lie on the circumference of a circle with centre O. Angle AOB = 130°. Find angle ACB.
Answer:
Angle ACB = 65° (angle at the centre is twice the angle at the circumference).
Question 1:
Points P, Q, and R lie on the circumference of a circle with centre O. Angle POR = 96°. Find angle PQR.
Question 2:
A, B, C, and D lie on the circumference of a circle. Angle BAD = 115°. Find angle BCD.
Question 3:
A tangent is drawn to a circle at point A. The tangent meets a chord AB such that the angle between the tangent and chord AB is 35°. Find the angle subtended by chord AB in the alternate segment.
§Formulas to memorise
The angle in a semicircle is 90°. — Any angle subtended by a diameter at the circumference is a right angle.
Angles in the same segment are equal. — Angles subtended by the same chord on the same side of the chord are equal.
Opposite angles in a cyclic quadrilateral sum to 180°. — A cyclic quadrilateral has all four vertices on the circumference.
Chord — a straight line joining two points on the circumference.
Tangent — a straight line that touches the circle at exactly one point.
Arc — a part of the circumference.
Segment — the region between a chord and the arc it cuts off.
Subtended — the angle "created by" or "standing on" an arc or chord.
Read the question — and identify what angle you need to find.
Label the diagram. — Mark the centre, any radii, tangents, and chords.
Worked example
Points A, B, and C lie on the circumference of a circle with centre O. Angle AOB = 130°. Find angle ACB.
Working:
⚠ Common mistakes
- ✗Forgetting to give reasons. Every angle you calculate must have a circle theorem or angle fact stated alongside it. "Angles in a triangle sum to 180°" counts as a reason.
- ✗Confusing the angle at the centre with the reflex angle. If the angle at the centre is marked as a reflex angle (greater than 180°), the corresponding angle at the circumference is half of that reflex angle, not half of the smaller angle.
- ✗Not spotting isosceles triangles. Two radii always form an isosceles triangle. Forgetting this leads to wrong base angles.
- ✗Mixing up the alternate segment theorem. The angle is between the tangent and the chord, not between the tangent and a different line. Trace the chord carefully.
- ✗Assuming a quadrilateral is cyclic. Only state that opposite angles sum to 180° if the question confirms all four vertices lie on the circumference.
✦ Exam tips
- →State the theorem by name — writing "angle at centre = 2 × angle at circumference" is sufficient and earns the reason mark.
- →Use two or more theorems. Higher-mark questions (4–5 marks) typically need a chain of reasoning involving multiple theorems.
- →Look for tangent + radius = 90° as your starting point whenever a tangent appears.
- →Mark equal angles on the diagram with matching arcs. This helps you visualise the solution.
- →Practise identifying the theorems from a diagram before calculating — this saves time under exam pressure.