Circle Theorems: Complete GCSE Maths Guide with Examples

Circle theorems are one of the most common topics on higher-tier GCSE Maths papers. They appear on AQA, Edexcel, and OCR exams every year, 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. The good news is that there are only eight theorems to learn, and once you know them, the questions become very predictable. This guide covers all eight, with clear explanations and worked examples.

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 formed at the centre is exactly twice the angle formed at the circumference. Both angles must be subtended by the same arc (the same two points on the circle).

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 (where the triangle’s longest side is the diameter) is a right angle. Put another way, if you draw a triangle inside a circle where one side is the diameter, the angle opposite the diameter is always exactly 90 degrees.

Example: A triangle is drawn 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 (that is, they stand on the same chord and are on the same side of it), those angles are equal. This is sometimes called the “same segment theorem” or “angles subtended by the same arc.”

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 is a four-sided shape where all four vertices lie on a circle. The opposite angles of a cyclic quadrilateral always add up to 180 degrees. This means if one angle is 110 degrees, the angle directly opposite it must be 70 degrees.

Example: In a 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 is a straight line that touches the circle at exactly one point. At that point, the tangent and the radius always meet at 90 degrees. This is one of the most frequently tested theorems because it unlocks right-angle triangle calculations within circle problems.

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

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

If you draw two tangent lines from a single point outside the circle, both tangents are exactly the same length (measured from the external point to the points 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. In other words, 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 the one students find hardest to spot. 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 an angle of 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

If you draw a line from the centre of a circle to the midpoint of a chord, that line is perpendicular to the chord. Equivalently, if a line from the centre is perpendicular to a chord, it cuts the chord exactly in half.

Example: A chord AB has length 10 cm. The perpendicular distance from the centre O to the chord meets it at point M. AM = MB = 5 cm. If the radius is 13 cm, you can find the 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 are subtended by arc BC. By Theorem 1 (angle at centre is twice angle at circumference):

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 in a cyclic quadrilateral 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 the centre O to an external point T. OA = 5 cm and OT = 13 cm. Find the length of the tangent AT.

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

AT² = OT² - OA² = 169 - 25 = 144

AT = 12 cm.

Tips for Remembering the Theorems

Group them by theme. Theorems 1, 2, and 3 are all about angles at the circumference. Theorem 4 is about cyclic quadrilaterals. Theorems 5, 6, and 7 involve tangents. Theorem 8 is about chords. Thinking in groups makes them easier to recall.

Use keywords. When you see a tangent in a diagram, immediately think “90 degrees to the radius” and “alternate segment.” When you see a quadrilateral inscribed in a circle, think “opposite angles add to 180.” When you see the diameter, think “angle in a semicircle is 90.”

Practise identifying theorems from diagrams. The hardest part of circle theorem questions is recognising which theorem applies. Work through as many past paper questions as you can, and for each one, write down which theorem you used and why. After 20 or 30 questions, spotting them becomes second nature.

Common Exam Mistakes

Not giving reasons. Circle theorem questions almost always say “give reasons for your answer.” You must name the theorem you are using. Writing “angle at centre is twice angle at circumference” earns you the reasoning mark. Just writing the numerical answer does not.

Confusing the alternate segment theorem with other angle rules. The alternate segment theorem specifically involves a tangent and a chord. Do not apply it to situations involving only chords or only secants.

Forgetting that the angle in a semicircle theorem requires a diameter. If the longest side of the triangle is not a diameter, this theorem does not apply. Always check that the line passes through the centre.

Misidentifying cyclic quadrilaterals. A quadrilateral is only cyclic if all four vertices lie on the circumference of the circle. If one vertex is at the centre, it is not a cyclic quadrilateral, and Theorem 4 does not apply.

Exam Technique

Always mark right angles on your diagram when you spot tangent-radius situations or angles in semicircles. Fill in as many angles as you can before you start writing your answer. Often, circle theorem questions require two or three steps, using different theorems in sequence.

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, so 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
• GCSE Further Maths revision guide — for students ready to go beyond Higher tier

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