Circle Theorems
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Revision notes for AQA GCSE Maths Circle Theorems. Open the guide for explanations and worked examples. Written against the AQA GCSE Maths (8300) specification, so the content matches what's examinable rather than general Maths background.

Circle Theorems

What you'll learn

  • Use the main GCSE circle angle rules confidently.
  • Spot hidden right-angled and isosceles triangles.
  • Chain facts together and give clear reasons for each angle or length.

Circle vocabulary

Definition

Circle vocabulary

  • The centre is the fixed middle point of the circle.
  • A radius is a line from the centre to the circumference.
  • The circumference is the outside edge of the circle.
  • A chord is a straight line joining two points on the circumference.
  • A tangent is a straight line that touches the circle at exactly one point.
  • An arc is part of the circumference.
  • A cyclic quadrilateral is a four-sided shape with all four vertices on the circumference.
  • An angle is subtended by a chord when its two arms join to the ends of that chord.

Key circle vocabulary: centre, radius, chord, tangent, arc and an angle subtended by a chord.

1. Equal radii make isosceles triangles

All radii in the same circle are equal. So if you join the centre to two points on the circumference, you often create an isosceles triangle, meaning a triangle with two equal sides.

Key Idea

Look for equal radii

If two sides of a triangle are radii of the same circle, the two base angles are equal.

Example

Finding an angle at the centre

A and B are on a circle with centre O. In triangle AOB, angle ABO is 42°. Find angle AOB.

Triangle AOB is isosceles because OA and OB are equal radii.

  1. OA and OB are both radii, so OA=OBOA = OBOA=OB.

  2. Triangle AOB is isosceles, so the base angles are equal: ∠OAB=42∘\angle OAB = 42^\circ∠OAB=42∘.

  3. Angles in a triangle add to 180°, so ∠AOB=180∘−42∘−42∘=96∘\angle AOB = 180^\circ - 42^\circ - 42^\circ = 96^\circ∠AOB=180∘−42∘−42∘=96∘.

2. A tangent meets a radius at 90°

Perpendicular means meeting at a right angle, 90°. The radius drawn to the point where a tangent touches the circle is perpendicular to the tangent.

Key Idea

Tangent-radius theorem

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

Example

Two tangents and a centre angle

A and C are points on a circle with centre O. Lines AB and CB are tangents, and angle ABC is 50°. Find angle OAC.

Two tangents from B create right angles with the radii at A and C, forming quadrilateral AOCB.

  1. OA meets tangent AB at A, so ∠OAB=90∘\angle OAB = 90^\circ∠OAB=90∘. OC meets tangent CB at C, so ∠OCB=90∘\angle OCB = 90^\circ∠OCB=90∘.

  2. Angles in quadrilateral AOCB add to 360°, so ∠AOC=360∘−90∘−90∘−50∘=130∘\angle AOC = 360^\circ - 90^\circ - 90^\circ - 50^\circ = 130^\circ∠AOC=360∘−90∘−90∘−50∘=130∘.

  3. OA and OC are radii, so OA=OCOA = OCOA=OC. Triangle AOC is isosceles.

  4. The base angles are equal, so ∠OAC=180∘−130∘2=25∘\angle OAC = \frac{180^\circ - 130^\circ}{2} = 25^\circ∠OAC=2180∘−130∘​=25∘.

Common Mistake

Wrong radius

The 90° angle is only between the tangent and the radius drawn to the exact point of contact.

3. Angle at the centre and angle at the circumference

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

Example

Using the centre-circumference rule

B and C are points on a circle with centre O. The angle BOC at the centre is 74°. A is another point on the circumference. Find angle BAC.

The centre angle BOC and circumference angle BAC both stand on the same chord BC.

  1. Both angles stand on chord BC.

  2. The angle at the centre is twice the angle at the circumference.

  3. Therefore ∠BAC=74∘2=37∘\angle BAC = \frac{74^\circ}{2} = 37^\circ∠BAC=274∘​=37∘.

Common Mistake

Check for reflex angles

A reflex angle is bigger than 180°. If the reflex centre angle is labelled, halve that reflex angle; if you need the smaller centre angle, subtract the reflex angle from 360° first.

4. Angles from chords

Two very useful chord facts are:

  • Angles in the same segment are equal.
  • Opposite angles in a cyclic quadrilateral add to 180°.
Example

Same segment and cyclic quadrilateral

A, B, C and D lie on a circle. C and D are in the same segment with chord AB. Angle ADB is 47°, and angle ADC is 78°. Find angles ACB and ABC.

Angles ADB and ACB stand on chord AB, and ABCD is a cyclic quadrilateral.

  1. Angles ADB and ACB both stand on chord AB.

  2. Angles in the same segment are equal, so ∠ACB=47∘\angle ACB = 47^\circ∠ACB=47∘.

  3. ABCD is a cyclic quadrilateral, so opposite angles add to 180°.

  4. Therefore ∠ABC=180∘−78∘=102∘\angle ABC = 180^\circ - 78^\circ = 102^\circ∠ABC=180∘−78∘=102∘.

5. Alternate segment theorem

The alternate segment theorem connects tangents and chords.

It says: the angle between a tangent and a chord is equal to the angle in the opposite segment.

Example

Tangent and chord angle

A, B and C are on a circle. A tangent touches the circle at C. Angle ABC is 58°, angle ACB is 71°, and x is the angle between the tangent and chord CB. Find x.

The tangent-chord angle x at C is linked to the angle in the opposite segment standing on chord CB.

  1. First find the missing angle in triangle ABC: ∠BAC=180∘−58∘−71∘=51∘\angle BAC = 180^\circ - 58^\circ - 71^\circ = 51^\circ∠BAC=180∘−58∘−71∘=51∘.

  2. The angle x is between the tangent and chord CB.

  3. By the alternate segment theorem, this equals the angle standing on chord CB in the opposite segment, which is angle BAC.

  4. So x=51x = 51x=51.

Tip

Match the chord carefully

If the tangent angle uses chord CB, look for the angle on the circumference made by chord CB. If it uses chord CA, look for the angle made by chord CA.

6. Length questions with tangents

Circle theorem questions can include ordinary geometry too. If you get a tangent and a radius, you often get a right-angled triangle, so you may need Pythagoras’ theorem.

Pythagoras’ theorem says that in a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.

Example

Finding a length using a tangent

AC is a tangent at A. O is the centre, and O, B and C lie on a straight line. OA is 6 cm and AC is 8 cm. Find BC.

The radius OA is perpendicular to tangent AC, creating right-angled triangle OAC for finding OC before subtracting OB.

  1. OA is a radius to the tangent at A, so triangle OAC is right-angled at A.

  2. Use Pythagoras’ theorem to find OC:

    OC2=62+82OC2=100OC=10\begin{aligned} OC^2 &= 6^2 + 8^2 \\ OC^2 &= 100 \\ OC &= 10 \end{aligned}OC2OC2OC​=62+82=100=10​

  3. OB is a radius, so OB is 6 cm.

  4. Since O, B and C are in a straight line, BC is 10 - 6 = 4 cm.

Exam technique

In the exam

  1. Mark all equal radii and all tangent-radius right angles on the diagram first.

  2. Identify the chord or arc involved before choosing a theorem.

  3. Write a reason after each calculation, such as “radii are equal” or “opposite angles in a cyclic quadrilateral add to 180°”.

  4. If a question has several marks, expect a chain of two or three facts, not just one theorem.

Self review

Check yourself

  • If OA and OB are radii and angle ABO is 35°, how could you find angle AOB?
  • Which theorem links a tangent-chord angle to an angle on the circumference?
  • What do opposite angles in a cyclic quadrilateral add to?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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Circle Theorems Revision Guide

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