GCSE Maths • Higher Tier
GCSE Circle Theorems Revision Guide
Learn the key angle, chord and tangent rules used in GCSE circle problems.
Foundation
Exam practice
Worked examples
What You'll Learn
Identify Circle Features
Recognise centres, radii, diameters, chords, arcs, segments, tangents and points of contact.
Use Centre Angles
Match angles on the same arc and use the 2:1 relationship correctly.
Recognise Equal Angles
Apply same-segment and alternate-segment rules only when their conditions are satisfied.
Apply Tangent Properties
Use perpendicular radii and equal tangent lengths to calculate angles and lengths.
Use Cyclic Quadrilaterals
Use opposite angles totalling 180 degrees.
Build Multi-Step Proofs
Combine theorems with triangle facts and algebra while giving valid reasons.
Key Rules
Lessons Module
Alternate Segment Theorem
Match the tangent-chord angle to the angle subtended by the same chord in the opposite segment.
Angles At The Centre
Use the 2:1 relationship for angles standing on the same arc.
Angles In The Same Segment
Recognise equal angles standing on the same chord in the same segment.
Tangent Theorems
Use perpendicular radius-tangent angles and equal tangents from a common point.
Core Subtopics
Worked Examples
Centre Angle and an Isosceles Triangle
QUESTION
A and B lie on a circle with centre O. Angle ACB at the circumference is 38 degrees and stands on chord AB. Find angle OAB.
- Given information: Angles ACB and AOB stand on the same chord AB. OA and OB are radii.
- Method: Find the centre angle, then use the equal radii and triangle angle sum.
- Angle AOB = 2 x 38 degrees = 76 degrees.
- OA = OB, so triangle AOB is isosceles..
- The equal base angles total 180 degrees - 76 degrees = 104 degrees.
- Angle OAB = 104 degrees / 2 = 52 degrees.
- Check: 52 + 52 + 76 = 180 degrees.
- Final answer: Angle OAB = 52 degrees.
- Independent answer check: The three angles in triangle AOB total 180 degrees and the base angles are equal.
- One common mistake: Do not give 76 degrees: that is the centre angle, not angle OAB.
- Common GCSE Mistakes
- Using the Wrong Arc or Chord
- Doubling the Wrong Angle
- Misusing Same-Segment Equality
- Missing the Point of Contact
- Using Adjacent Cyclic Angles
- Trusting the Drawing
- Exam Tips
- Mark centres, equal radii, chords, tangents and right angles first.
- Trace the relevant arc or chord.
- Write a reason beside every angle statement.
- Combine theorems with triangle and straight-line angle facts.
- Keep algebra exact until the unknown is found.
- Check angle sums and supplementary relationships.
Ready to Practice?
Printable Worksheet
Practise theorem recognition, calculations and reasons.
Interactive Quiz
Test all core circle-theorem rules.
Exam-Style Questions
Apply several rules with algebra and proof.
Watch the Circle Theorems Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Frequently Asked Questions
They cover centre and circumference angles, same-segment angles, semicircles, cyclic quadrilaterals, radius-tangent angles, equal tangents and alternate segments.
When both angles stand on the same arc.
They stand on the same chord and intercept the same arc from the same side.
It touches once; the radius there is perpendicular, and two tangents from one external point have equal lengths.
Look for a tangent meeting a chord; that angle equals the circumference angle subtended by the chord in the opposite segment.
State each relationship and give its named theorem or standard angle reason.
