Revising circle theorems the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Circle theorems GCSE” on the site for the complete guide.
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Key facts to remember
- 1Angle at centre = 2 × angle at circumference (same arc).
- 2Angles in the same segment are equal.
- 3Angle in a semicircle = 90° (angle subtended by a diameter).
- 4Opposite angles in a cyclic quadrilateral sum to 180°.
- 5Tangent to a circle is perpendicular to the radius at the point of contact.
- 6Tangents from an external point are equal in length.
- 7Alternate segment theorem: angle between tangent and chord = angle in alternate segment.
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Worked examples
Example 1O is the centre of a circle. Angle AOB = 112°. Find angle ACB, where C is a point on the major arc.
Working
- Angle at centre = 2 × angle at circumference
- Angle ACB = 112° ÷ 2 = 56°
Answer56°
Example 2ABCD is a cyclic quadrilateral. Angle A = 78° and angle B = 105°. Find angles C and D.
Working
- Opposite angles in a cyclic quadrilateral sum to 180°
- Angle C = 180° − 78° = 102°
- Angle D = 180° − 105° = 75°
AnswerC = 102°, D = 75°
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Common mistakes
✗Confusing angle at the centre with angle at the circumference — the centre angle is always double.
✗Not stating the circle theorem used — examiners require the reason.
✗Assuming all angles in the same circle are equal (they must be subtended by the same arc).
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Exam tips
✓Learn all seven theorems by name and always write the full theorem as your reason.
✓Mark the centre O clearly and identify which arc the angles are subtended by before applying theorems.
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