Circle Theorems GCSE: All 8 Rules Explained

Circle theorems are a set of rules about angles and lengths inside and around a circle. At GCSE maths you need to know eight key theorems, be able to identify which one applies in a diagram, state it by name, and use it to find missing angles — often giving reasons in your working.

Why do circle theorems matter at GCSE?

Circle theorem questions almost always ask you to find an angle and give a reason. The reason must name the theorem — for example, "angle at the centre is twice the angle at the circumference." Without the reason, even a correct numerical answer can score no marks for reasoning.

The golden habit is: underline or annotate the diagram, identify which theorem applies, write the theorem name as your reason, then calculate.

What are the eight circle theorems?

Here is a summary of all eight theorems you need for GCSE.

# | Theorem | Key statement
1 | Angle at the centre | The angle at the centre is twice the angle at the circumference, when both are subtended by the same arc
2 | Angles in the same segment | Angles subtended by the same arc on the same side of a chord are equal
3 | Angle in a semicircle | The angle in a semicircle is always 90°
4 | Cyclic quadrilateral | Opposite angles in a cyclic quadrilateral add up to 180°
5 | Tangent–radius | A tangent to a circle is perpendicular to the radius at the point of contact
6 | Two tangents from a point | Two tangents drawn from the same external point are equal in length
7 | Alternate segment theorem | The angle between a tangent and a chord equals the angle in the alternate segment
8 | Perpendicular from centre to chord | The perpendicular from the centre to a chord bisects the chord

How does the angle-at-the-centre theorem work?

Imagine two points A and B on a circle. They subtend an angle at the centre O and also an angle at any point P on the major arc.

Theorem 1: Angle AOB = 2 × angle APB.

Worked example: The angle at the centre is 110°. What is the angle at the circumference subtended by the same arc?

Angle at circumference = 110° ÷ 2 = 55°

A common trap: the angle at the centre must be the reflex version when P is on the minor arc. In that case, reflex AOB = 2 × angle APB, so you use the reflex angle, not 110°.

Theorem 3 (angle in a semicircle) is a special case: when AB is a diameter, the angle at the centre is 180°, so the angle at the circumference is 180° ÷ 2 = 90°. Any angle inscribed in a semicircle is always a right angle.

How do you use the cyclic quadrilateral theorem?

A cyclic quadrilateral is a four-sided shape whose four vertices all lie on a circle.

Theorem 4: Opposite angles in a cyclic quadrilateral add up to 180°.

So if angles are labelled A, B, C, D going around the quadrilateral:
A + C = 180° and B + D = 180°.

Worked example: In cyclic quadrilateral PQRS, angle P = 73° and angle R = ?

Angle R = 180° − 73° = 107°

Reason: opposite angles in a cyclic quadrilateral sum to 180°.

What do the tangent theorems tell us?

A tangent is a straight line that touches the circle at exactly one point.

Theorem 5: The angle between the tangent and the radius drawn to the point of contact is always 90°. This means if you know the tangent direction and the radius direction, the angle between them is a right angle — use Pythagoras or trigonometry to find missing lengths.

Theorem 6: If two tangents are drawn from an external point T to a circle (touching at points A and B), then TA = TB. The triangle formed is isosceles.

Worked example: A tangent from external point T touches the circle at A. The radius OA = 5 cm and OT = 13 cm. Find TA.

Triangle OAT has a right angle at A (tangent ⊥ radius).
TA² = OT² − OA² = 169 − 25 = 144
TA = 12 cm

What is the alternate segment theorem?

The alternate segment theorem (Theorem 7) is one of the trickiest at GCSE. It states: the angle between a tangent to a circle and a chord drawn from the point of tangency equals the angle subtended by that chord in the alternate segment (the segment on the other side of the chord).

In plain terms: draw a tangent at point P, draw a chord PA. The angle between the tangent and chord PA equals the angle that chord PA makes at any point on the arc on the opposite side.

Worked example: The angle between tangent and chord PA is 48°. What is the angle in the alternate segment?

Angle in the alternate segment = 48°

They are equal — that is exactly what the theorem states.

How do you write circle theorem reasons in an exam?

Examiners expect a specific form for the reason. Use these exact phrasings:

"Angle at the centre is twice the angle at the circumference (same arc)"

"Angles in the same segment are equal"

"Angle in a semicircle is 90°"

"Opposite angles in a cyclic quadrilateral sum to 180°"

"Tangent is perpendicular to radius at the point of contact"

"Two tangents from an external point are equal in length"

"Alternate segment theorem"

"Perpendicular from the centre bisects the chord"

Abbreviations such as "∠ at centre = 2 × ∠ at circumference" are generally accepted, but write the reason in full if in doubt.

Frequently asked questions

Do I need to prove the circle theorems in a GCSE exam?

No. GCSE questions ask you to apply the theorems to find angles or lengths, and to state which theorem you used. You will not be asked to prove the theorems from scratch at GCSE, though understanding why they work helps you remember them.

What is the most commonly tested circle theorem?

The angle at the centre (Theorem 1) and the cyclic quadrilateral (Theorem 4) appear most frequently at GCSE. The alternate segment theorem is the one students find hardest and often appears on higher-tier papers. Practise all eight, but prioritise these three.

How many reasons do I need to give if I use two circle theorems in one question?

You must state a reason for each step where you use a theorem. If you use two theorems in sequence, write two reasons — one alongside each calculation. Missing a reason loses a mark even if the numbers are correct.

What is a cyclic quadrilateral?

A cyclic quadrilateral is any four-sided polygon whose four corners all lie on the circumference of a single circle. Not every quadrilateral is cyclic — for example, a general quadrilateral drawn freehand is unlikely to have all four vertices on a circle unless it has been specifically constructed that way.

For Socratic GCSE geometry practice, see aitutors.me.