Circle Theorems: The 8 Rules for GCSE Maths, With Examples
Quick answer: There are eight circle theorems at GCSE. The most used are: the angle at the centre is twice the angle at the circumference, the angle in a semicircle is 90°, opposite angles in a cyclic quadrilateral add to 180°, and a tangent meets a radius at 90°. In the exam, you must name the theorem you used, not just give the angle.
The eight theorems
Angle at the centre: The angle at the centre is twice the angle at the circumference, from the same arc
Angle in a semicircle: An angle drawn from a diameter to the circumference is 90°
Same segment: Angles in the same segment are equal
Cyclic quadrilateral: Opposite angles add up to 180°
Tangent and radius: A tangent meets a radius at 90°
Two tangents: Tangents from the same outside point are equal in length
Chord bisector: The perpendicular from the centre to a chord cuts the chord in half
Alternate segment: The angle between a tangent and a chord equals the angle in the alternate segment
Worked examples
1. Angle at the centre
The angle at the circumference is 35°. Find the angle at the centre from the same arc.
70°. Reason: the angle at the centre is twice the angle at the circumference.
2. Angle in a semicircle
A triangle is drawn in a semicircle. One angle is 28°. Find the other non-right angle.
The angle at the circumference is 90°, so 180 − 90 − 28 = 62°.
3. Cyclic quadrilateral
One angle of a cyclic quadrilateral is 110°. Find the opposite angle.
180 − 110 = 70°. Reason: opposite angles in a cyclic quadrilateral add up to 180°.
4. Tangent and radius
A circle has radius 5 cm. A point P is 13 cm from the centre. Find the length of the tangent from P.
The tangent meets the radius at 90°, so use Pythagoras: √(13² − 5²) = √144 = 12 cm.
5. Alternate segment
The angle between a tangent and a chord is 50°. Find the angle in the alternate segment.
50°. Reason: the alternate segment theorem.
How to write reasons
Examiners look for the theorem in words. "Angle at centre is twice angle at circumference" earns the mark. "Circle rule" doesn't. Write one reason for every step.
This tip is MMS editorial guidance.
Common mistakes
Using the angle at the centre rule with different arcs. Both angles must stand on the same arc.
Forgetting the 90° in a semicircle. Look for a diameter.
Mixing up the alternate segment. It's the segment on the other side of the chord.
When extra support might help
Support is worth considering when circle questions keep losing marks for missing reasons.
A tutor can't guarantee a grade, but can drill the reasons until they're automatic. Math Make Smart offers online maths tutoring in the UK, and you can book a free trial lesson. Related: our arc length and sector guide and GCSE trigonometry guide.
Frequently asked questions
How many circle theorems are there for GCSE?
Eight are usually taught.
Are circle theorems on the Foundation paper?
They're generally Higher tier content. Check your exam board's specification.
Do I need to write the reason?
Yes. Reasons usually carry marks.