Master all 8 circle theorems for GCSE Maths Higher tier. Angles in a semicircle, cyclic quadrilaterals, tangent properties, and more — with diagrams and worked examples.
Circle theorems are a group of rules about angles and lines relating to circles. They're a Higher tier topic that regularly appears on GCSE papers for 3–6 marks. Questions ask you to find missing angles AND state the reason — both must be correct for full marks.
The angle at the centre of a circle is twice the angle at the circumference, when both angles are subtended by the same arc.
The angle in a semicircle is always 90°. If one side of a triangle inscribed in a circle is a diameter, the angle opposite to it (on the circumference) is always a right angle.
Angles subtended by the same chord (from the same side) are equal. If two inscribed angles look at the same arc, they're the same size.
In a cyclic quadrilateral (all four vertices on a circle), opposite angles add up to 180°. This means the opposite angles are supplementary.
Four key rules involve tangents (lines that touch the circle at exactly one point).
No — you need to apply them to find angles, and state the theorem name as your reason. You don't need to formally prove the theorems themselves.
The angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle subtending the same chord on the opposite side. It's one of the trickier theorems to spot.
Typically 3–6 marks. Usually 1 mark for the correct angle and 1 mark for the correct reason. In multi-step problems, each step has its own marks, so always show your working and state each theorem used.
Exam Ladder has hundreds of adaptive practice questions on this topic — with instant AI explanations when you get stuck.
Start Free TrialNo credit card required