Circle Theorems: Complete GCSE Maths Guide with Examples

Circle theorems show up on higher-tier GCSE Maths papers every single year. AQA, Edexcel, OCR — they all love them. Typically worth 4 to 6 marks per question. This guide is part of our GCSE Maths revision tips series — start there for a full overview of revision strategies.

Here’s the good news: there are only eight theorems. Learn them properly, and the questions become predictable. The pattern in mark scheme analyses is consistent: students who know these eight cold rarely drop marks, while students who half-know them lose marks everywhere.

The 8 Circle Theorems

1. The Angle at the Centre Is Twice the Angle at the Circumference

When two points on a circle are joined to both the centre and a point on the circumference, the angle at the centre is exactly twice the angle at the circumference. Both angles must be subtended by the same arc.

Example: If the angle at the circumference is 35 degrees, the angle at the centre subtended by the same arc is 70 degrees.

2. The Angle in a Semicircle Is 90 Degrees

Any angle inscribed in a semicircle is a right angle. Draw a triangle inside a circle where one side is the diameter. The angle opposite the diameter? Always exactly 90 degrees.

Example: A triangle sits inside a circle with one side as the diameter. The angle at the circumference opposite the diameter is 90 degrees. If one of the other angles is 40 degrees, the remaining angle is 50 degrees (since 90 + 40 + 50 = 180).

3. Angles in the Same Segment Are Equal

If two or more angles at the circumference are subtended by the same arc — meaning they stand on the same chord and sit on the same side of it — those angles are equal. Sometimes called the “same segment theorem.”

Example: Two triangles share a common chord AB. Points C and D are both on the same arc. Angle ACB = angle ADB.

4. Opposite Angles in a Cyclic Quadrilateral Sum to 180 Degrees

A cyclic quadrilateral has all four vertices on a circle. Opposite angles always add up to 180 degrees. One angle is 110 degrees? The one directly opposite must be 70 degrees.

Example: In cyclic quadrilateral ABCD, angle A = 105 degrees and angle C = 75 degrees. Check: 105 + 75 = 180 degrees. If angle B = 82 degrees, then angle D = 98 degrees.

5. A Tangent to a Circle Is Perpendicular to the Radius at the Point of Contact

A tangent touches the circle at exactly one point. At that point, tangent and radius meet at 90 degrees. This theorem gets tested constantly because it opens up right-angle triangle calculations.

Example: A tangent meets a circle at point P. The radius OP makes 90 degrees with the tangent at P. If a line from centre O to an external point makes 35 degrees with radius OP, the angle between that line and the tangent is 90 - 35 = 55 degrees.

6. Two Tangents from an External Point Are Equal in Length

Draw two tangent lines from a single point outside the circle. Both tangents are exactly the same length, measured from the external point to where they touch the circle. The line from the external point to the centre also bisects the angle between the two tangents.

Example: From point T outside a circle, two tangents are drawn to points A and B on the circle. TA = TB. If TA = 12 cm, then TB = 12 cm.

7. The Alternate Segment Theorem

The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment. The angle that the chord makes with the tangent on one side equals the angle subtended by that chord on the opposite arc.

This theorem is consistently the hardest of the eight to spot in mock conditions. Look for a tangent meeting a chord at the edge of the circle. The angle between them equals the angle at the circumference on the far side of the chord.

Example: A tangent at point P makes 50 degrees with chord PQ. The angle at the circumference in the alternate segment (angle PRQ, where R is on the major arc) is also 50 degrees.

8. The Perpendicular from the Centre to a Chord Bisects the Chord

Draw a line from the centre of a circle to the midpoint of a chord. That line is perpendicular to the chord. Flip it round: if a line from the centre is perpendicular to a chord, it cuts the chord exactly in half.

Example: Chord AB has length 10 cm. The perpendicular from centre O meets the chord at point M. AM = MB = 5 cm. If the radius is 13 cm, find perpendicular distance OM using Pythagoras: OM² + 5² = 13², so OM² = 144, and OM = 12 cm.

Worked Exam-Style Examples

Example 1: Finding a Missing Angle

Points A, B, C, and D lie on a circle with centre O. Angle BOC = 124 degrees. Find angle BAC.

Solution: Angle BOC is at the centre. Angle BAC is at the circumference. Both subtended by arc BC. By Theorem 1:

Angle BAC = 124 / 2 = 62 degrees.

Example 2: Cyclic Quadrilateral

PQRS is a cyclic quadrilateral. Angle P = 3x + 10 degrees. Angle R = 2x + 20 degrees. Find x and both angles.

Solution: Opposite angles sum to 180:

(3x + 10) + (2x + 20) = 180

5x + 30 = 180

5x = 150

x = 30

Angle P = 3(30) + 10 = 100 degrees. Angle R = 2(30) + 20 = 80 degrees. Check: 100 + 80 = 180 degrees.

Example 3: Tangent and Radius

A tangent at point A on a circle meets a line from centre O to external point T. OA = 5 cm and OT = 13 cm. Find the length of tangent AT.

Solution: By Theorem 5, angle OAT = 90 degrees. Triangle OAT is right-angled. Pythagoras:

AT² = OT² - OA² = 169 - 25 = 144

AT = 12 cm.

Tips for Remembering the Theorems

Group them by theme. Theorems 1, 2, and 3 deal with angles at the circumference. Theorem 4 covers cyclic quadrilaterals. Theorems 5, 6, and 7 involve tangents. Theorem 8 is about chords. Groups stick better than lists.

Use keywords. See a tangent? Think “90 degrees to the radius” and “alternate segment.” See a quadrilateral inscribed in a circle? Think “opposite angles add to 180.” See the diameter? Think “angle in a semicircle is 90.”

Practise identifying theorems from diagrams. The hardest part is recognising which theorem applies. Work through past paper questions. For each one, write down which theorem you used and why. After 20 or 30 questions, spotting them becomes automatic.

Common Exam Mistakes

The most common dropped-mark error: finding the correct angle but not naming the theorem. The question says “give reasons for your answer” — that means you must write something like “angle at centre is twice angle at circumference.” Just writing “62 degrees” gets you the calculation mark but loses you the reasoning mark. This shows up in roughly half of all scripts. Free marks, thrown away.

Confusing the alternate segment theorem with other angle rules. This theorem specifically involves a tangent and a chord. Don’t apply it when you’ve only got chords or secants.

Forgetting that the semicircle theorem requires a diameter. If the longest side of the triangle doesn’t pass through the centre, this theorem doesn’t apply. Check every time.

Misidentifying cyclic quadrilaterals. All four vertices must lie on the circumference. If one vertex is at the centre, it’s not a cyclic quadrilateral, and Theorem 4 doesn’t apply.

Exam Technique

Mark right angles on your diagram the moment you spot tangent-radius situations or angles in semicircles. Fill in as many angles as you can before writing your answer. Circle theorem questions often need two or three steps, chaining different theorems together.

A reliable strategy when a circle theorems question looks impenetrable: start by labelling everything you know, even if it seems obvious. The path to the answer often only becomes clear once the obvious bits are written down.

If a question asks you to prove a result, state each theorem explicitly and show every intermediate angle. Examiners award marks for each correct step and reason — be thorough rather than rushing to the final answer.

Related GCSE Maths Guides

• Geometry and measures guide — angles, shapes, transformations, and Pythagoras
• Pythagoras’ theorem guide — formula, proof, and 3D applications

How to Use This Guide

Don’t just read this once and hope for the best. Print it out or keep it open while you work through past papers. Every time you hit a circle theorem question, come back here and check which theorem applies. After a few sessions, you’ll stop needing the guide — the patterns will be in your head. If you’re still getting questions wrong after 15-20 attempts, focus on the specific theorem that’s catching you out. That’s where the marks are hiding.

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