GCSE Maths Circle Theorems Notes
Circle theorems are a set of angle rules for circles. Learn each rule and, crucially, the reason to quote — reasons carry marks in the exam.
Key concepts
- Angle at the centre
- The angle at the centre is twice the angle at the circumference from the same arc.
- Angle in a semicircle
- The angle in a semicircle is always 90°.
- Cyclic quadrilateral
- Opposite angles of a cyclic quadrilateral sum to 180°.
- Tangent facts
- A tangent meets a radius at 90°, and two tangents from a point are equal in length.
- Angles in the same segment (from the same arc) are equal.
- The perpendicular from the centre to a chord bisects the chord.
- The alternate segment theorem links a tangent–chord angle to the angle in the alternate segment.
Worked example
A, B and C sit on a circle. The angle at the centre AOC is 140°. Find angle ABC at the circumference.
- 1The angle at the centre is twice the angle at the circumference from the same arc.
- 2So angle ABC = 140° ÷ 2.
- 3Calculate: 70°.
Answer: 70°
Common mistakes
✗ Not stating the theorem used.
✓ Always quote the rule, e.g. 'angle at centre is twice angle at circumference', for the reasoning mark.
✗ Mixing up 'same segment' with 'angle at the centre'.
✓ Same segment → angles equal; centre vs circumference → double/half.
✗ Forgetting the tangent–radius right angle.
✓ Where a tangent touches, the radius is perpendicular (90°).
Quick check
Try these, then reveal the answers.
The angle in a semicircle is?Reveal answer
90°
Opposite angles of a cyclic quadrilateral sum to?Reveal answer
180°
A tangent meets a radius at what angle?Reveal answer
90°
Exam tips
- Memorise the reason wording for each theorem — the marks are for the reason as much as the answer.
- Mark equal angles and right angles on the diagram as you spot them.
- Look for isosceles triangles formed by two radii; their base angles are equal.
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Frequently asked questions
Do I need to give reasons for circle theorem answers?+
Yes — most marks are for quoting the correct theorem, so always state the rule you used.
What is the alternate segment theorem?+
The angle between a tangent and a chord equals the angle in the alternate segment made by that chord.
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