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GCSE Maths / Edexcel

Circle theorems

Recognise the main GCSE circle theorems from a diagram, use them to find missing angles, and explain the angle reason clearly.

Geometry and MeasuresHigherGrades 6 to 9Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Circle theorems

Pearson Edexcel GCSE Maths Higher geometry G10: apply and prove the standard circle theorems concerning angles, radii, tangents and chords.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Circle Theorems feel hard because the diagram is often crowded. Start by naming what you can see: centre, radius, diameter, chord, circumference and tangent.

A radius goes from the centre to the circle. A chord joins two points on the circle. A diameter is a chord through the centre. A tangent touches the circle at one point.

Most circle theorems are about the same chord or the same arc. If two angles stand on the same chord, they are connected.

Angle in a semicircle: if a triangle is drawn on a diameter and the third point is on the circle, the angle at the third point is 90 degrees.

Angle at the centre: the angle at the centre is twice the angle at the circumference standing on the same arc.

Angles in the same segment are equal. This means two angles at the circumference standing on the same chord are equal.

Opposite angles in a cyclic quadrilateral add to 180 degrees. Cyclic means all four vertices are on the circumference.

A radius meets a tangent at 90 degrees at the point of contact.

Alternate segment theorem: the angle between a tangent and a chord equals the angle in the opposite segment. This is usually the theorem students find hardest, so look for the tangent first.

For Edexcel marks, do not just write the number. Write the theorem or angle reason, especially in Higher questions.

Key ruleIdentify the circle feature first, then use the matching theorem: semicircle 90, centre double, same segment equal, cyclic opposite angles 180, radius to tangent 90.

Diagram guide

diameterradiuschordtangentsectorarc is part of the circumference
Start with circle vocabularyBefore using a theorem, identify whether the diagram shows a diameter, chord, tangent, radius or cyclic quadrilateral.
diameter90if the longest side is a diameterthe angle on the circle is 90 degrees
Angle in a semicircleIf the base of the triangle is a diameter, the angle on the circumference is 90 degrees.
2xxOsame arc AB: centre angle is doubleangle at centre = 2 x angle at circumference
Centre angle theoremThe angle at the centre is twice the angle at the circumference when both stand on the same arc.
xxsame chord AB gives equal anglesangles in the same segment are equal
Same segment theoremAngles standing on the same chord in the same segment are equal.
a180 - aall four corners lie on the circleopposite angles add to 180 degrees
Cyclic quadrilateralOpposite angles in a cyclic quadrilateral add to 180 degrees.
radiustangent90radius to tangent point is perpendicularlook for the point where the tangent touches
Tangent and radiusThe radius to the point where a tangent touches the circle meets the tangent at 90 degrees.
xxtangentangle between tangent and chordequals the angle in the opposite segment
Alternate segment theoremThe angle between a tangent and chord equals the angle in the opposite segment.

Worked examples

Angle in a semicircle

AB is a diameter of a circle. C is on the circumference. Find angle ACB.

diameter90if the longest side is a diameterthe angle on the circle is 90 degrees
Example: semicircleThe angle standing on a diameter is always 90 degrees.
  1. AB is a diameter.
  2. C lies on the circumference.
  3. Angle in a semicircle is 90 degrees.

Answer: 90 degrees

Angle at the centre

Angle ACB at the circumference is 34 degrees. A, B and C are on the circle and O is the centre. Find angle AOB.

2xxOsame arc AB: centre angle is doubleangle at centre = 2 x angle at circumference
Example: centre angleThe centre angle and circumference angle stand on the same arc AB.
  1. Angle at the centre is twice the angle at the circumference.
  2. Angle AOB = 2 x 34.
  3. Angle AOB = 68 degrees.

Answer: 68 degrees

Cyclic quadrilateral

ABCD is a cyclic quadrilateral. Angle A is 112 degrees. Find angle C.

a180 - aall four corners lie on the circleopposite angles add to 180 degrees
Example: opposite anglesOpposite angles in a cyclic quadrilateral add to 180 degrees.
  1. A and C are opposite angles.
  2. Opposite angles in a cyclic quadrilateral add to 180 degrees.
  3. Angle C = 180 - 112 = 68 degrees.

Answer: 68 degrees

Tangent and radius

A tangent touches a circle at T. OT is a radius. What is the angle between OT and the tangent?

radiustangent90radius to tangent point is perpendicularlook for the point where the tangent touches
Example: tangent at TThe right angle is at the point where the tangent touches the circle.
  1. OT is a radius to the point of contact.
  2. A radius meets a tangent at 90 degrees.
  3. So the angle between OT and the tangent is 90 degrees.

Answer: 90 degrees

Alternate segment theorem

The angle between a tangent and chord AB is 58 degrees. Find the angle in the opposite segment standing on chord AB.

xxtangentangle between tangent and chordequals the angle in the opposite segment
Example: alternate segmentMatch the tangent-chord angle with the angle on the opposite side of the chord.
  1. The tangent and chord make an angle of 58 degrees.
  2. By the alternate segment theorem, this equals the angle in the opposite segment.
  3. The required angle is 58 degrees.

Answer: 58 degrees

Common mistakes

  • Using a theorem before checking whether the line is a diameter, chord or tangent.
  • Thinking every triangle inside a circle has a 90-degree angle. It must stand on a diameter.
  • Doubling the wrong angle in the centre theorem.
  • Treating a cyclic quadrilateral like any quadrilateral without checking all four vertices are on the circle.
  • Using tangent-radius 90 degrees at the wrong point. The right angle is at the point of contact.
  • Forgetting to give the circle theorem reason in a Higher exam question.

Quick exercise

Try these before moving to the exam-style questions.

  1. AB is a diameter and C is on the circumference. What is angle ACB?
    diameter90if the longest side is a diameterthe angle on the circle is 90 degrees
    Quick check: semicircleAngle in a semicircle is 90 degrees.
  2. An angle at the circumference is 41 degrees. Find the angle at the centre standing on the same arc.
    2xxOsame arc AB: centre angle is doubleangle at centre = 2 x angle at circumference
    Quick check: centre angleCentre angle is double.
  3. Two angles in the same segment stand on the same chord. One is 67 degrees. Find the other.
    xxsame chord AB gives equal anglesangles in the same segment are equal
    Quick check: same segmentAngles in the same segment are equal.
  4. Opposite angles in a cyclic quadrilateral are 104 degrees and x. Find x.
    a180 - aall four corners lie on the circleopposite angles add to 180 degrees
    Quick check: cyclic quadrilateralOpposite angles add to 180 degrees.
  5. A radius meets a tangent at the point of contact. What angle do they make?
    radiustangent90radius to tangent point is perpendicularlook for the point where the tangent touches
    Quick check: tangentRadius to tangent is perpendicular.
  6. The angle between a tangent and chord is 49 degrees. What is the angle in the opposite segment?
    xxtangentangle between tangent and chordequals the angle in the opposite segment
    Quick check: alternate segmentThe matching angle in the opposite segment is equal.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleHigherNon-calculator3 marks

AB is a diameter of a circle. C lies on the circumference. Angle CAB is 38 degrees. Work out angle CBA. Give a reason.

diameter90if the longest side is a diameterthe angle on the circle is 90 degrees
Question diagram: semicircle triangleFirst find the right angle, then use the triangle angle sum.
circle theoremsemicircletriangle angle sum
Standard exam styleHigherNon-calculator3 marks

A, B and C are points on a circle with centre O. Angle ACB is 47 degrees. Work out angle AOB, giving a reason.

2xxOsame arc AB: centre angle is doubleangle at centre = 2 x angle at circumference
Question diagram: centre theoremThe two angles stand on the same arc AB.
circle theoremangle at centrehigher geometry
ChallengeHigherNon-calculator5 marks

ABCD is a cyclic quadrilateral. Angle A is 106 degrees. A tangent at B makes an angle of 54 degrees with chord BC. Work out angle C and state the circle theorem used for the tangent angle.

xxtangentangle between tangent and chordequals the angle in the opposite segment
Question diagram: mixed circle theoremsUse cyclic opposite angles, then identify the tangent-chord theorem.
circle theoremcyclic quadrilateralalternate segment theorem