IGCSE Maths Circle Theorems Notes
Circle theorems are rules about angles and lines in circles. Quoting the correct theorem earns reasoning marks.
Key concepts
- Angle at the centre
- The angle at the centre is twice the angle at the circumference on the same arc.
- Angle in a semicircle
- The angle in a semicircle is 90°.
- Cyclic quadrilateral
- Opposite angles of a cyclic quadrilateral sum to 180°.
Key formulas
Centre vs circumference
angle at centre = 2 × angle at circumference
Cyclic quadrilateral
opposite angles = 180°
Worked example
The angle at the centre is 120°. Find the angle at the circumference on the same arc.
- 1The centre angle is twice the circumference angle.
- 2Divide by 2: 120 ÷ 2.
- 3State the circumference angle.
Answer: 60°
Common mistakes
✗ Not quoting the theorem used.
✓ Name the theorem to earn the reasoning mark.
✗ Confusing the tangent–radius right angle.
✓ A tangent meets a radius at 90°.
✗ Mixing up centre and circumference angles.
✓ The centre angle is the larger, doubled one.
Quick check
Try these, then reveal the answers.
Angle in a semicircle?Reveal answer
90°
Opposite angles in a cyclic quad sum to?Reveal answer
180°
Tangent meets radius at?Reveal answer
90°
Exam tips
- Always state the reason, e.g. 'angle at centre = 2 × angle at circumference'.
- Mark equal angles and right angles on the diagram.
- Look for isosceles triangles formed by two radii.
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Frequently asked questions
Do I need to give reasons in circle theorem questions?+
Yes — marks are awarded for naming the correct theorem, not just the numerical answer.
What angle does a tangent make with a radius?+
A tangent is always perpendicular (90°) to the radius at the point of contact.
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