GCSE Maths → Geometry → Circle Theorems

Circle Theorems GCSE Maths: Step-by-Step Guide

Circle theorems are an important GCSE Higher maths topic. They are rules about angles and lines in circles. Many students find them difficult because the diagrams can look similar, but each question becomes easier when you learn how to spot the key features.

In exams, you usually need to identify the correct theorem, calculate missing angles, and sometimes give a written reason for your answer.

Video explanation

A short Worthing Maths Tutor video explanation for circle theorems GCSE maths can be embedded here later to improve student engagement and time on page.

What students need to know first

Basic angle facts, including angles on a straight line.
Angles in a triangle add to 180°.
Angles around a point add to 360°.
Vertically opposite angles are equal.
Basic circle vocabulary: radius, diameter, chord, tangent, arc, and circumference.

Exam tip: Circle theorem questions often combine circle rules with ordinary angle facts. Do not stop after spotting one theorem.

Main GCSE circle theorems

1. Angle at the centre is twice the angle at the circumference

If two angles stand on the same arc, the angle at the centre is twice the angle at the circumference.

Example 1: Centre and circumference

If the angle at the circumference is 34°, then the angle at the centre is double.

2 × 34° = 68°

So the angle at the centre is 68°.

2. Angles in the same segment are equal

Angles in the same segment are equal when they stand on the same chord or arc.

Example 2: Same segment

If one angle in the same segment is 52°, the other angle in the same segment is also 52°.

x = 52°

3. Angle in a semicircle is 90°

If a triangle is drawn inside a semicircle and one side is the diameter, the angle opposite the diameter is always 90°.

Example 3: Semicircle theorem

If a triangle sits on the diameter of a circle, the angle at the circumference is:

90°

4. Opposite angles in a cyclic quadrilateral add to 180°

A cyclic quadrilateral is a four-sided shape where all four corners lie on the circumference of a circle.

Example 4: Cyclic quadrilateral

If one angle in a cyclic quadrilateral is 110°, the opposite angle is:

180° - 110° = 70°

5. Tangent and radius meet at 90°

A tangent touches the circle at exactly one point. The radius drawn to that point meets the tangent at a right angle.

Example 5: Tangent and radius

If a radius meets a tangent at the point of contact, the angle between them is:

90°

6. Tangents from the same external point are equal

If two tangents are drawn from the same point outside the circle, the tangent lengths are equal.

Example 6: Equal tangents

If one tangent from an external point is 8 cm, the other tangent from the same point is also:

8 cm

7. Alternate segment theorem

The angle between a tangent and a chord is equal to the angle in the opposite segment.

Example 7: Alternate segment theorem

If the angle between the tangent and chord is 41°, then the angle in the opposite segment is also:

41°

Common mistake: The alternate segment theorem is often confused with the tangent-radius rule. First check whether the angle is between a tangent and a chord, not between a tangent and a radius.

How to answer circle theorem questions

Look for key words or diagram features: tangent, radius, chord, diameter, cyclic quadrilateral.
Mark any obvious 90° angles first.
Look for angles standing on the same arc or chord.
Use normal angle facts such as triangle sum or straight line angles.
If the question asks for a reason, write the theorem name clearly.

Exam tip: In GCSE exams, marks are often awarded for giving the correct reason, not just the correct angle. Write the theorem name when asked.

Common mistakes in circle theorems

Using a theorem that does not match the diagram.
Forgetting that a tangent and radius meet at 90°.
Mixing up “angle at the centre” and “angle at the circumference”.
Forgetting that opposite angles in a cyclic quadrilateral add to 180°.
Not giving a reason when the question asks students to explain.
Trying to memorise words without practising diagrams.

Practice questions

Try these before checking the answers.

The angle at the circumference is 38°. Find the angle at the centre.
The angle at the centre is 96°. Find the angle at the circumference on the same arc.
One angle in a cyclic quadrilateral is 115°. Find the opposite angle.
A radius meets a tangent. What is the angle between them?
An angle in a semicircle is marked x. What is x?
Two angles are in the same segment. One is 47°. Find the other.
The angle between a tangent and a chord is 63°. Find the angle in the opposite segment.

Answers

76°
48°
65°
90°
90°
47°
63°

Circle theorems FAQ

What are circle theorems?

Circle theorems are angle rules involving circles, chords, tangents, radii, diameters, and cyclic quadrilaterals.

Are circle theorems on GCSE Higher maths?

Yes. Circle theorems are usually tested on GCSE Higher maths and often appear in multi-step geometry questions.

Why do students struggle with circle theorems?

Students often struggle because they memorise the words but cannot recognise which theorem applies from the diagram.

How should I revise circle theorems?

Revise circle theorems by drawing diagrams, labelling angles, naming each theorem, and practising exam-style questions with reasons.

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