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 answers using correct mathematical language. This revision guide lists every theorem you need, works through exam-style examples, flags the mistakes students make most often, and gives you practice questions to test yourself. For a wider overview of geometry topics, check our complete GCSE Maths formulas list.

What Are Circle Theorems?

Circle theorems are a set of rules that describe the relationship between angles, chords, tangents, and other lines associated with circles. They are used to find unknown angles when geometric figures are drawn inside or around a circle.

The Theorems You Must Know

1. The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.
2. The angle in a semicircle is 90°. Any angle subtended by a diameter at the circumference is a right angle.
3. Angles in the same segment are equal. Angles subtended by the same chord on the same side of the chord are equal.
4. Opposite angles in a cyclic quadrilateral sum to 180°. A cyclic quadrilateral has all four vertices on the circumference.
5. The tangent to a circle is perpendicular to the radius at the point of contact. The angle between a tangent and a radius is 90°.
6. Two tangents drawn from an external point are equal in length.
7. The alternate segment theorem. The angle between a tangent and a chord equals the angle in the alternate segment.
8. The perpendicular from the centre to a chord bisects the chord.

Key Vocabulary

• 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.

Step-by-Step Method

1. Read the question and identify what angle you need to find.
2. Label the diagram. Mark the centre, any radii, tangents, and chords.
3. Look for isosceles triangles — any triangle with two sides that are radii is isosceles.
4. Identify which theorem(s) apply. Often you need more than one theorem in the same question.
5. Calculate the angle and write a clear reason for every step. Examiners require reasons — a bare number without justification will not earn full marks.
6. Check your answer. Do the angles in each triangle sum to 180°? Do opposite angles in any cyclic quadrilateral sum to 180°?

Worked Example 1 — Higher Level

Question: Points A, B, and C lie on the circumference of a circle with centre O. Angle AOB = 130°. Find angle ACB.

Working:

Step 1 — Identify the theorem. Both angle AOB (at the centre) and angle ACB (at the circumference) are subtended by the same arc AB.

Step 2 — Apply the theorem: the angle at the centre is twice the angle at the circumference.

Angle ACB = 130° ÷ 2 = 65°

Answer: Angle ACB = 65° (angle at the centre is twice the angle at the circumference).

Worked Example 2 — Higher Level

Question: A, B, C, and D are points on the circumference of a circle. The tangent at A meets the chord BC extended at point T. Angle TAB = 48°. Angle ABC = 72°. Find angle ADC.

Working:

Step 1 — Use the alternate segment theorem. Angle TAB = angle ACB (the angle in the alternate segment). So angle ACB = 48°.

Step 2 — In triangle ABC: angle BAC = 180° − 72° − 48° = 60°.

Step 3 — ABCD is a cyclic quadrilateral. Opposite angles sum to 180°. Angle ADC + angle ABC = 180° Angle ADC = 180° − 72° = 108°

Answer: Angle ADC = 108°.

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.

Practice Questions

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.

Question 4: Points A, B, and C lie on a circle. AC is a diameter. Angle BAC = 37°. Find angle ABC.

Question 5: Two tangents are drawn from an external point T to a circle with centre O, touching the circle at A and B. Angle ATB = 50°. Find angle AOB.

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Related Topics

• Circle Theorems blog post — more examples and visual guides.
• Angles in Polygons — angle rules that often combine with circle theorems.
• Arc Length and Sector Area — another circle-based topic.
• Congruence and Similarity — proofs that can involve circles.

Summary

Circle theorems test your ability to spot relationships between angles in circle diagrams. The eight key theorems cover angles at the centre and circumference, angles in semicircles, cyclic quadrilaterals, tangent properties, and the alternate segment theorem. Success depends on careful labelling, identifying isosceles triangles formed by radii, and always stating the theorem you are using. These questions are Higher only, typically worth 3–5 marks, and require fully reasoned answers. Practise reading diagrams and chaining multiple theorems together — that is the skill the examiner is really testing.