Circle theorems are one of the most searched IGCSE maths topics because they appear in geometry papers and many students find them hard. You need to name each theorem, state it correctly, and apply it to find angles (and sometimes lengths).

Key Circle Theorems for IGCSE

• Angle at centre = 2 × angle at circumference (same arc).
• Angle in a semicircle = 90°.
• Angles in the same segment are equal.
• Cyclic quadrilateral – Opposite angles add to 180°.
• Tangent and radius – Tangent perpendicular to radius at point of contact.
• Alternate segment – Angle between tangent and chord = angle in alternate segment.
• Tangents from a point – Two tangents from same point are equal in length.

Always label the diagram and state the theorem you use so the examiner can award method marks.

How to Tackle Circle Theorem Questions

• Mark known angles and equal angles/sides on the diagram.
• Identify which theorem gives you a new angle or length.
• Work step by step; sometimes you need two theorems in sequence.
• Give reasons (e.g. “angle in same segment”) in your working.

Practise with past paper circle theorem questions so you recognise the setups quickly.

Worked Example and Common Errors

Example: Find angle x where the angle at the centre is 140°. Angle at centre = 2 × angle at circumference (same arc), so x = 140°/2 = 70°. Always state the theorem you use so the examiner can award method marks. Common errors: Using the wrong theorem (check which angles or lengths you have and what the theorem requires); mixing up “angle at centre” and “angle at circumference”; mislabelling the diagram. Draw the diagram clearly and mark known angles as you go. Circle theorems often combine (e.g. angle in semicircle then angles in same segment)—work step by step.

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