Circle theorems, explained

By The STEM-Tutor team · Updated October 2026

Circle theorems appear on GCSE Maths Higher papers, and exam questions usually ask you to give reasons. Here are all eight in plain English.

The eight circle theorems

1. The angle at the centre is twice the angle at the circumference, when both stand on the same arc.
2. The angle in a semicircle is 90°. An angle made at the circumference by a diameter is a right angle.
3. Angles in the same segment are equal. Angles at the circumference standing on the same chord are equal.
4. Opposite angles in a cyclic quadrilateral add up to 180°.
5. A tangent meets a radius at 90° at the point where it touches the circle.
6. Two tangents from the same external point are equal in length.
7. The perpendicular from the centre to a chord bisects the chord.
8. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

How to spot which theorem to use

• A diameter in the diagram? Look for a right angle in a semicircle.
• A tangent? Look for a 90° angle with a radius, or the alternate segment theorem.
• Four points on the circle forming a quadrilateral? Opposite angles add to 180°.
• Lines from the centre and the circumference to the same arc? The centre angle is double.

Giving reasons for full marks

Most circle theorem questions say "give reasons for each stage of your working". Use the standard wording, for example:

• "The angle at the centre is twice the angle at the circumference."
• "Opposite angles in a cyclic quadrilateral add up to 180°."
• "Angles in a triangle add up to 180°."
• "Base angles of an isosceles triangle are equal." Two radii make an isosceles triangle.

Common mistakes

• Using the centre-angle theorem when the angles stand on different arcs.
• Calling a quadrilateral cyclic when one vertex is not on the circle.
• Writing the right answer without reasons, which loses marks.

How to practise

1. Learn the eight theorems with a sketch for each.
2. Practise exam questions and write a reason for every step.
3. Look for isosceles triangles made by radii; they unlock many questions.

Worked examples with reasons

Example 1: angle at the centre

Angle AOB at the centre is 130°. Find angle APB at the circumference, on the same arc.

• APB = 130° ÷ 2 = 65°.
• Reason: the angle at the centre is twice the angle at the circumference.

Example 2: isosceles triangles from radii

O is the centre, A and B are on the circle, and angle AOB = 110°. Find angle OAB.

• OA = OB, so triangle OAB is isosceles. Reason: radii are equal.
• OAB = (180° − 110°) ÷ 2 = 35°. Reason: angles in a triangle add up to 180° and base angles of an isosceles triangle are equal.

Example 3: cyclic quadrilateral

ABCD is a cyclic quadrilateral with angle A = 105°. Find angle C.

• C = 180° − 105° = 75°.
• Reason: opposite angles in a cyclic quadrilateral add up to 180°.

Example 4: two tangents

Tangents from point P touch the circle at A and B. Angle APB = 50°. Find angle AOB.

• Angles OAP and OBP are both 90°. Reason: a tangent meets a radius at 90°.
• OAPB is a quadrilateral, so AOB = 360° − 90° − 90° − 50° = 130°. Reason: angles in a quadrilateral add up to 360°.

Example 5: alternate segment theorem

A tangent at T makes an angle of 70° with chord TA. Find the angle TBA, where B is on the circle in the other segment.

• TBA = 70°. Reason: the angle between a tangent and a chord equals the angle in the alternate segment.

Why the centre-angle theorem is true

Higher papers sometimes ask you to prove a circle theorem. The classic proof:

1. Draw the line from P through the centre O, splitting the diagram into two triangles, OAP and OBP.
2. Each triangle is isosceles because OA = OP = OB (radii). Call the base angles x and y.
3. The exterior angle of triangle OAP at O is 2x, and of triangle OBP is 2y.
4. So angle AOB = 2x + 2y = 2(x + y) = 2 × angle APB.

A routine for any circle question

1. Mark every radius and look for isosceles triangles.
2. Mark every tangent with a right angle at the radius.
3. Look for a diameter (angle in a semicircle) or a four-point shape on the circle (cyclic quadrilateral).
4. Write each angle you find on the diagram with its reason.

Practice questions

Try each one before opening the answer.

Example 6: combining two theorems

O is the centre. Angle ABC = 40°, where B is on the circumference and A and C are on the circle. Find angle OAC.

• Angle AOC = 2 × 40° = 80°. Reason: the angle at the centre is twice the angle at the circumference.
• Triangle OAC is isosceles because OA = OC (radii), so OAC = (180° − 80°) ÷ 2 = 50°. Reason: base angles of an isosceles triangle are equal, and angles in a triangle add up to 180°.

Frequently asked questions

Do I need to learn circle theorem proofs?

Higher papers can ask you to prove a theorem, most often the angle at the centre or the cyclic quadrilateral theorem. Learn the proof above and practise writing it with reasons.

Are circle theorems on Foundation papers?

Circle theorems are a Higher tier topic. Foundation students should still know the parts of a circle: radius, diameter, chord, tangent, arc, sector and segment.

How many marks are circle theorem questions worth?

Usually three to five marks, with marks for each correct angle and for each correct reason. Writing reasons is often the difference between a grade 7 and a grade 9.