GCSE Maths · Topic guide
GCSE Maths circle theorems
Circle theorem questions usually give a diagram: you must state the correct theorem, give a short reason, then calculate. Higher tier includes multi-step angle chasing; foundation may use fewer theorems but still expects clear vocabulary ('angle at centre', 'angles in same segment'). Show the angle you're using and avoid assuming lines are parallel or tangents unless marked or stated.
Common mistakes in exams
- Confusing angle at centre (2× angle at circumference) with equal angles in the same segment.
- Using the wrong arc when two angles subtend different chords.
- Forgetting that a tangent is perpendicular to the radius at the point of contact.
- Not giving a geometric reason when the mark scheme asks for one.
Worked examples & mini quiz
GCSE Maths: Circle theorems
Key facts: angle in a semicircle is 90°; angle at centre = 2 × angle at circumference; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; tangent ⊥ radius.
Worked examples
Example 1
Angle at centre
If angle at circumference = 35°, angle at centre on the same arc = 70°.
Example 2
Cyclic quadrilateral
Opposite angles sum to 180°. If one angle is 110°, the opposite is 70°.
Mini quiz (3 questions)
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1. An angle inscribed in a semicircle is:
- A.90°
- B.45°
- C.180°
- D.Half the arc length
Show answer
Correct: 90°
Angle in a semicircle theorem.
2. The angle at the centre is:
- A.Twice the angle at the circumference on the same arc
- B.Equal to the angle at the circumference
- C.Always 180°
- D.Half the angle at the circumference
Show answer
Correct: Twice the angle at the circumference on the same arc
Central angle = 2 × inscribed angle on the same arc.
3. A tangent to a circle meets a radius at the point of contact at:
- A.90°
- B.0°
- C.45°
- D.It varies
Show answer
Correct: 90°
Tangent ⊥ radius.
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