Circle theorems: all 8 rules with examples

TutorAura guide · Updated 9 November 2026

Circle theorems are rules about angles and lines in circles. GCSE Higher and IGCSE Extended students need to know them, use them to find missing angles, and state the theorem as a reason for each step. There are eight to learn; most questions combine one or two of them with basic angle facts (angles in a triangle add to 180°, angles on a straight line add to 180°).

Key takeaways

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 up to 180°.
A tangent meets a radius at 90°.

The eight theorems

Theorem	In short
1. Angle at the centre	The angle at the centre is twice the angle at the circumference, standing on the same arc.
2. Angle in a semicircle	An angle in a semicircle (from a diameter) is 90°.
3. Angles in the same segment	Angles standing on the same chord, in the same segment, are equal.
4. Cyclic quadrilateral	Opposite angles add up to 180°.
5. Tangent and radius	A tangent is perpendicular (90°) to the radius at the point of contact.
6. Two tangents	Tangents from the same outside point are equal in length.
7. Alternate segment	The angle between a tangent and a chord equals the angle in the alternate segment.
8. Perpendicular to a chord	A perpendicular from the centre to a chord bisects the chord.

Worked examples

Angle at the centre: the angle at the centre is 110°. The angle at the circumference on the same arc is $110 \div 2 = 55°$ .
Cyclic quadrilateral: one angle is 75°. The opposite angle is $180 - 75 = 105°$ .
Tangent: a tangent touches the circle at $A$ , and $O$ is the centre. Angle $OAT = 90°$ ; if angle $AOT = 50°$ , the third angle of triangle $OAT$ is $180 - 90 - 50 = 40°$ .

How to spot each theorem

Two lines meeting at the centre and two at the edge, from the same two points: angle at the centre.
A triangle with one side a diameter: angle in a semicircle.
A "bow tie" shape: angles in the same segment.
Four points on the circle joined up: cyclic quadrilateral.
A line touching the circle at one point: tangent rules.
A tangent plus a chord at the same point: alternate segment.

Giving reasons

Most marks need a reason as well as the angle. Write the theorem in words:

"Angle at the centre is twice the angle at the circumference."
"Opposite angles in a cyclic quadrilateral add up to 180°."
"Tangent meets radius at 90°."

Also use basic facts as reasons: "angles in a triangle add up to 180°", "base angles of an isosceles triangle are equal" (two radii make an isosceles triangle).

Isosceles triangles hiding in circles

Any triangle with two radii as sides is isosceles, so its base angles are equal. Look for the centre $O$ joined to two points on the circle. This is the missing step in many circle theorem questions.

Common mistakes

Using the angle at the centre rule when the angles stand on different arcs.
Treating a quadrilateral as cyclic when one vertex isn't on the circle (or is the centre).
Forgetting the reasons.

Practise angle facts with our angles in polygons questions and circles questions, and use the circle calculator for lengths and areas.

Frequently asked questions

How many circle theorems are there?

Eight are usually taught at GCSE and IGCSE: angle at the centre, angle in a semicircle, angles in the same segment, cyclic quadrilaterals, tangent and radius, two tangents, alternate segment, and the perpendicular from the centre to a chord.

What is the alternate segment theorem?

The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment (on the other side of the chord).

Are circle theorems on GCSE Foundation?

Proving and using circle theorems is Higher tier at GCSE; IGCSE includes them in the Extended syllabus.

Sources

GCSE mathematics: subject content and assessment objectives. Department for Education.
Cambridge IGCSE Mathematics 0580 syllabus. Cambridge International.

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