GCSE Maths · Geometry and measures

Circle Theorems Exam Questions and Worked Answers

Circle theorems are exact angle and length relationships created by radii, chords, tangents and points on a circle. In an exam, identify the matching relationship, calculate with ordinary angle facts and state the theorem as your reason.

Edexcel · AQA · OCRHigher tier15 original questions

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Read the objects before recalling rules

What you need to know about circle theorems

A circle has a centre and a curved boundary called the circumference. A radius joins the centre to the circumference. A chord joins two points on the circumference. A tangent is a straight line that touches the circle at exactly one point. Circle theorems describe relationships made by these concrete objects.

Picture → relationship → angle reason

Name the lines in the diagram first

The same line segment can play a different role depending on its endpoints. Find the centre and touching point before choosing an angle fact.

Radiuscentre → circumferenceall radii in one circle are equal

Chordcircumference → circumferencea diameter is a chord through the centre

Tangentone touching pointit is perpendicular to the radius there

Radius, chord and tangent. Use the labelled relationships, not visual measurement.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Use the centre–circumference relationship

What the problem asks: Find one angle when the same arc forms an angle at the centre and an angle at the circumference.

How to solve it: The centre angle is twice the circumference angle on the same arc.

$\angle AOC=2\angle ABC$

Use a diameter

What the problem asks: Find an angle in a triangle whose base is a diameter and whose third vertex lies on the circumference.

How to solve it: The angle subtended by a diameter at the circumference is 90°.

$\angle ABC=90^\circ$

Use the same segment

What the problem asks: Compare two angles at the circumference that stand on the same chord and lie in the same segment.

How to solve it: Angles subtended by the same chord in the same segment are equal.

$\angle ABC=\angle ADC$

Use a cyclic quadrilateral

What the problem asks: Find an angle in a four-sided shape whose four vertices lie on the circumference.

How to solve it: Opposite angles in a cyclic quadrilateral add to 180°.

$\angle ABC+\angle ADC=180^\circ$

Use a tangent

What the problem asks: Relate a tangent to a radius, chord or another tangent.

How to solve it: A tangent is perpendicular to the radius at contact; tangent lengths from one external point are equal; the tangent–chord angle equals the angle in the alternate segment.

$OT\perp\text{ tangent},\qquad PA=PB$

Use a chord and the centre

What the problem asks: Find a chord length or prove two chord pieces are equal when a perpendicular from the centre is shown.

How to solve it: A perpendicular from the centre to a chord bisects the chord, creating two equal lengths and right-angled triangles.

$OM\perp AB\quad\Rightarrow\quad AM=MB$

A reliable routine

Method for multi-step circle theorem questions

Use this routine when a diagram combines one or more circle theorems with triangle, straight-line or isosceles-angle facts. It works by turning each known relationship into one justified local step.

Mark the centre, radii, chords, diameter, tangent and any four points on the circumference.
Write the circle theorem that directly connects the given angle or length to the unknown region.
Calculate one new fact at a time, also using equal radii, angles in a triangle or angles on a straight line when needed.
Write a reason beside every important angle statement.
Check that triangle totals are 180° and that the final angle size fits its position in the diagram.

Check: Do not write only ‘circle theorem’. Name the relationship, for example ‘opposite angles in a cyclic quadrilateral sum to 180°’.

Fully worked

Circle theorems GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Angle at the centre

2 marks

Question

Angle $AOC=116^\circ$ , where $O$ is the centre. Point $B$ is on the circumference on the opposite arc. Find angle $ABC$ .

Angle at the centre and circumference. Use the labelled relationships, not visual measurement.

The centre angle is twice the circumference angle on the same arc.

$\angle ABC=116^\circ\div2$

$\boxed{\angle ABC=58^\circ}$

Example 2

Angle in a semicircle

2 marks

Question

$AC$ is a diameter and $B$ is on the circumference. Find angle $ABC$ .

Angle in a semicircle. Use the labelled relationships, not visual measurement.

A diameter subtends a right angle at the circumference.

$\boxed{\angle ABC=90^\circ}$

Example 3

Angles in the same segment

2 marks

Question

Angles $ABC$ and $ADC$ stand on the same chord $AC$ in the same segment. If $\angle ABC=43^\circ$ , find $\angle ADC$ .

Angles standing on chord AC in the same segment. Use the labelled relationships, not visual measurement.

Angles in the same segment are equal.

$\boxed{\angle ADC=43^\circ}$

Example 4

Opposite cyclic angles

2 marks

Question

$ABCD$ is a cyclic quadrilateral. Angle $ABC=112^\circ$ . Find angle $ADC$ .

Cyclic quadrilateral. Use the labelled relationships, not visual measurement.

Opposite angles in a cyclic quadrilateral sum to $180^\circ$ .

$\angle ADC=180^\circ-112^\circ$

$\boxed{\angle ADC=68^\circ}$

Example 5

Radius and tangent

3 marks

Question

A tangent touches a circle at $T$ , and $OT$ is a radius. A line through $O$ makes an angle of $34^\circ$ with $OT$ . Find its acute angle with the tangent.

Radius perpendicular to a tangent. Use the labelled relationships, not visual measurement.

The radius and tangent meet at $90^\circ$ .

$90^\circ-34^\circ=\boxed{56^\circ}$

Example 6

Alternate segment theorem

2 marks

Question

The angle between a tangent at $A$ and chord $AB$ is $51^\circ$ . Point $C$ lies on the opposite arc. Find angle $ACB$ .

Tangent and alternate segment. Use the labelled relationships, not visual measurement.

The angle between a tangent and chord equals the angle in the alternate segment subtended by that chord.

$\boxed{\angle ACB=51^\circ}$

Example 7

Perpendicular from the centre to a chord

4 marks

Question

A circle has radius $13$ cm. The perpendicular distance from its centre to chord $AB$ is $5$ cm. Find the length of $AB$ .

Perpendicular from the centre to a chord. Use the labelled relationships, not visual measurement.

The perpendicular from the centre bisects the chord. Let $M$ be the midpoint of $AB$ .

In right-angled triangle $OMA$ ,

$AM^2+5^2=13^2$

$AM^2=169-25=144$

$AM=12$

Therefore

$AB=2(12)=\boxed{24\text{ cm}}$

Example 8

Prove the centre-angle theorem

4 marks

Question

Points $A$ , $B$ and $C$ lie on a circle with centre $O$ . Prove that the angle at the centre standing on arc $AC$ is twice $\angle ABC$ .

Construction for the centre-angle proof. Use the labelled relationships, not visual measurement.

Join $O$ to $B$ . Let $\angle ABO=x$ and $\angle OBC=y$ .

Because $OA=OB$ , triangle $AOB$ is isosceles.

$\angle AOB=180^\circ-2x$

Because $OB=OC$ , triangle $BOC$ is isosceles.

$\angle BOC=180^\circ-2y$

Angles around $O$ total $360^\circ$ , so the remaining angle standing on arc $AC$ is

$360^\circ-(180^\circ-2x)-(180^\circ-2y)=2x+2y$

But $\angle ABC=x+y$ . Therefore

$\boxed{\angle AOC=2\angle ABC}$

15 original questions · total 45 marks

Circle theorems GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 55 minutes · write a named angle reason for every theorem step and do not measure the diagrams · answers start collapsed

Halve a centre angle

2 marks

An angle at the centre is $146^\circ$ . Find the angle at the circumference on the same arc.

Show worked answer

$146^\circ\div2=\boxed{73^\circ}$

The centre angle is twice the circumference angle.

Double a circumference angle

2 marks

An angle at the circumference is $37^\circ$ . Find the angle at the centre on the same arc.

Show worked answer

$2(37^\circ)=\boxed{74^\circ}$

Use a semicircle with a triangle

3 marks

$AC$ is a diameter. Point $B$ lies on the circumference and $\angle BAC=28^\circ$ . Find $\angle BCA$ .

Triangle drawn on diameter AC. Use the labelled relationships, not visual measurement.

Show worked answer

$\angle ABC=90^\circ$

because the angle in a semicircle is a right angle.

$\angle BCA=180^\circ-90^\circ-28^\circ$

$\boxed{\angle BCA=62^\circ}$

Use the same chord

2 marks

Angles $ABC$ and $ADC$ lie in the same segment and stand on chord $AC$ . If $\angle ABC=67^\circ$ , find $\angle ADC$ .

Two angles standing on chord AC. Use the labelled relationships, not visual measurement.

Show worked answer

Angles in the same segment are equal.

$\boxed{\angle ADC=67^\circ}$

Find an opposite cyclic angle

2 marks

$ABCD$ is cyclic and $\angle ABC=97^\circ$ . Find $\angle ADC$ .

Cyclic quadrilateral ABCD. Use the labelled relationships, not visual measurement.

Show worked answer

$\angle ADC=180^\circ-97^\circ$

$\boxed{\angle ADC=83^\circ}$

Form and solve a cyclic equation

4 marks

Opposite angles of a cyclic quadrilateral are $(3x+8)^\circ$ and $(5x-4)^\circ$ . Find $x$ .

Show worked answer

Opposite cyclic angles sum to $180^\circ$ .

$3x+8+5x-4=180$

$8x+4=180$

$8x=176$

$\boxed{x=22}$

Use a tangent and radius

2 marks

A radius meets a tangent at $T$ . Find the angle between them.

Radius OT meeting a tangent at T. Use the labelled relationships, not visual measurement.

Show worked answer

A tangent is perpendicular to the radius at the point of contact.

$\boxed{90^\circ}$

Use equal tangents

2 marks

Tangents $PA$ and $PB$ are drawn from the same external point $P$ . If $PA=7.6$ cm, find $PB$ .

Two tangents from external point P. Use the labelled relationships, not visual measurement.

Show worked answer

Tangents from the same external point are equal in length.

$\boxed{PB=7.6\text{ cm}}$

Use the alternate segment

2 marks

The angle between tangent $AT$ and chord $AB$ is $39^\circ$ . Find the angle subtended by chord $AB$ at the opposite circumference.

Tangent, chord and alternate-segment angle. Use the labelled relationships, not visual measurement.

Show worked answer

By the alternate segment theorem, the two angles are equal.

$\boxed{39^\circ}$

Find a chord length

4 marks

A circle has radius $10$ cm. The perpendicular distance from the centre to a chord is $6$ cm. Find the chord length.

Show worked answer

The perpendicular bisects the chord. For half the chord,

$h^2+6^2=10^2$

$h^2=64$

$h=8$

$\text{chord}=2h=\boxed{16\text{ cm}}$

Use equal radii

3 marks

$O$ is the centre and $A,B$ lie on the circle. Angle $AOB=104^\circ$ . Find angle $OAB$ .

Show worked answer

Triangle $AOB$ is isosceles because $OA=OB$ .

$\angle OAB=\frac{180^\circ-104^\circ}{2}$

$\boxed{\angle OAB=38^\circ}$

Combine two circle facts

4 marks

$AC$ is a diameter and $D$ lies on the circumference. A tangent at $A$ makes an angle of $32^\circ$ with chord $AD$ . Find angles $ACD$ and $ADC$ .

Diameter AC, tangent at A and chord AD. Use the labelled relationships, not visual measurement.

Show worked answer

By the alternate segment theorem,

$\angle ACD=32^\circ$

Because $AC$ is a diameter,

$\angle ADC=90^\circ$

$\boxed{\angle ACD=32^\circ,\quad \angle ADC=90^\circ}$

Explain a theorem

3 marks

Explain why the angle subtended by a diameter at the circumference is $90^\circ$ .

Show worked answer

The diameter forms an angle of $180^\circ$ at the centre.

The angle at the centre is twice the angle at the circumference on the same arc.

$180^\circ\div2=90^\circ$

Therefore an angle in a semicircle is $\boxed{90^\circ}$ .

Prove angles in the same segment are equal

4 marks

Angles $ABC$ and $ADC$ stand on the same chord $AC$ . Use the centre-angle theorem to prove that they are equal.

Two circumference angles standing on chord AC. Use the labelled relationships, not visual measurement.

Show worked answer

Both circumference angles stand on the same chord $AC$ and the same centre angle $AOC$ .

By the centre-angle theorem,

$\angle AOC=2\angle ABC$

and

$\angle AOC=2\angle ADC$

Therefore

$2\angle ABC=2\angle ADC$

$\boxed{\angle ABC=\angle ADC}$

Solve a multi-step tangent problem

6 marks

$O$ is the centre of a circle. $A$ and $B$ lie on the circumference, and $\angle AOB=140^\circ$ . A tangent is drawn at $A$ .

(a) Find $\angle OAB$ .

(b) Find the acute angle between the tangent and chord $AB$ .

Radius, chord and tangent for a multi-step problem. Use the labelled relationships, not visual measurement.

Show worked answer

Triangle $AOB$ is isosceles because $OA=OB$ .

$\angle OAB=\frac{180^\circ-140^\circ}{2}=20^\circ$

The radius $OA$ is perpendicular to the tangent.

$\text{tangent--chord angle}=90^\circ-20^\circ=70^\circ$

By the alternate segment theorem, the angle subtended by $AB$ at the opposite circumference is also $70^\circ$ .

$\boxed{20^\circ,\ 70^\circ,\ 70^\circ}$

Examiner-style feedback

Common circle theorems mistakes

Using the visible shape instead of the labels

Circle theorem diagrams are often not drawn accurately. Use centres, equal radii, chords and tangent marks.

Applying a rule to different arcs

Centre and circumference angles must stand on the same arc; same-segment angles must stand on the same chord.

Forgetting ordinary angle facts

Many questions also need equal base angles, a 180° triangle total or a straight-line angle.

Giving no reason

Write the full theorem relationship. An unsupported angle may lose a reasoning mark.

Using the wrong centre angle

Check whether the circumference angle stands on the minor or major arc. The matching centre angle may be reflex, so compare the endpoints of both angles.

Matching the wrong alternate-segment angle

Use the angle between the tangent and the named chord, then match the circumference angle standing on that same chord.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Name the circle objects first.
Match the given and unknown to one theorem.
Use one justified angle step at a time.
State the exact theorem and check ordinary angle totals.

Quick answers

Circle theorems FAQ

How many circle theorems do I need for GCSE?

Courses commonly group the required relationships into seven families covering centre and circumference angles, semicircles, same segments, cyclic quadrilaterals, tangents and chords.

Are circle theorem diagrams drawn to scale?

Do not assume so. Use the labels and theorem conditions rather than measuring or judging the picture.

What does subtended mean?

An arc or chord subtends an angle when lines from its two endpoints meet at the angle’s vertex.

How do I earn proof marks?

State each new angle or equality and give its precise reason, including ordinary triangle and straight-line facts.

Build connected skills

What to revise next

Coordinate circles

Equation of a circle

Connect radius, coordinates and tangent gradients.

Revise equation of a circle

Right triangles

Trigonometry

Find missing sides and angles in geometric problems.

Revise trigonometry

Direction

Bearings

Combine angle facts with three-figure directions.

Revise bearings

Content standards

Curriculum and rights review

Reviewed 5 September 2026 against DfE content G10, Pearson Edexcel G10, AQA G10 and OCR J560 sections 8.05b–h. All questions, proofs and diagrams are original.

Official specification references