Every circle theorem you need, how to spot them in exam questions, and how to write proof answers that get full marks.
Circle theorems are one of the most reliably tested topics in GCSE Higher Maths, and they're also one of the most learnable — there are exactly eight theorems, they're all visually distinct, and once you know them properly you can identify which one applies within seconds of seeing a diagram. The problem is most students learn the statements without understanding why they're true, which means they can't apply them to unfamiliar diagrams or write proper proofs.
This guide covers all eight theorems, explains the logic behind each one, shows you how to spot them in exam diagrams, and covers the proof questions that appear at Higher tier.
Circle theorem questions use specific vocabulary. If you don't know these terms precisely, you'll misread diagrams and lose marks on proofs.
If angle AOB is at the centre and angle ACB is at the circumference, both subtended by arc AB, then angle AOB = 2 × angle ACB.
This is the most fundamental circle theorem — several others are actually derived from it. The key visual cue is seeing both a central angle and a circumference angle opening up to the same two points on the circle. The point C can be anywhere on the major arc and the relationship holds.
A common exam trap: the angle at the centre might be a reflex angle. If the circumference angle is obtuse (say 130°), the central angle on the same side is 260° (the reflex). Don't automatically assume the central angle is the smaller one.
Any two angles subtended by the same chord, from the same side of the chord, are equal — regardless of where on the arc the vertex sits.
This follows directly from Theorem 1. Both angles are half the central angle subtended by the same arc, so they must be equal to each other. In diagrams, you'll see two angles that both "open up" to the same chord from the same side. They look different positions but they're identical in size.
If AB is a diameter, then any angle ACB where C is on the circumference equals exactly 90°.
This is a special case of Theorem 1. The angle at the centre subtended by a diameter is 180° (a straight line). The angle at the circumference is therefore half of 180° = 90°. This theorem is tested constantly — whenever you see a triangle inside a circle with one side being the diameter, one angle is 90°.
If you see a diameter in a circle diagram, immediately mark the angle at the circumference opposite it as 90°. This unlocks the rest of the question — you now have a right-angled triangle and can use Pythagoras or trigonometry.
In a quadrilateral with all four vertices on a circle, angle A + angle C = 180° and angle B + angle D = 180°.
Again this derives from Theorem 1. Opposite angles are subtended by arcs that together make the full circle (360°). Each angle is half its arc, so the two opposite angles together are half of 360° = 180°.
The exam application: if you know three angles of a cyclic quadrilateral, you can find the fourth. Or if you need to prove something is a cyclic quadrilateral, show that opposite angles sum to 180°.
Where a tangent meets the circle, the angle between the tangent and the radius is exactly 90°.
This one is visually very obvious once you know it — the tangent and radius form a right angle. In exam diagrams, this right angle is sometimes marked explicitly, sometimes not. If you see a tangent line and a radius drawn to the point of tangency, mark that right angle yourself before doing anything else.
If two tangent lines are drawn from a point P outside the circle, touching the circle at A and B, then PA = PB.
This follows from the congruence of the two right-angled triangles formed (both share the hypotenuse from P to the centre, both have a radius as one leg, both have a right angle). So the tangent lengths must be equal. This theorem is used in problems involving perimeters of shapes that include tangent segments.
If a chord AB meets a tangent at A, the angle between the tangent and the chord equals the angle subtended by the chord in the opposite segment.
This is the theorem students find hardest to spot. The visual cue is a tangent line touching the circle, with a chord drawn from the point of tangency. There are two angles to consider: the angle between the tangent and the chord (on one side), and the angle subtended by that same chord from the opposite segment of the circle. Those two angles are equal.
In exams, this theorem often appears in multi-step questions where you first use another theorem to find an angle, then use the alternate segment theorem to find a second angle. Practise identifying which angle is in the "alternate segment" — draw it explicitly on the diagram if needed.
If a line from the centre meets a chord at 90°, it cuts the chord exactly in half.
This one is less about angles and more about lengths. It's used in problems where you need to find a chord length or the distance from the centre to a chord, using Pythagoras' theorem. The perpendicular from the centre creates two right-angled triangles with the radius as hypotenuse, and those triangles are congruent — hence the chord is bisected.
PaperPlus generates GCSE circle theorem questions at Higher tier — including multi-step and proof questions — with full mark schemes for AQA, Edexcel and OCR.
Start Practising Free →At Higher tier, you'll sometimes be asked to prove a circle theorem rather than just apply it. These questions are worth several marks and many students score zero because they don't know what a proof actually requires.
A proof requires: stating what you know (given information), applying logical steps with reasons, and reaching the conclusion. Every geometric statement needs a reason — you can't just write "angle = 40°" without saying why.
Valid reasons for circle theorem proofs include: "angle at centre is twice angle at circumference", "angles in the same segment are equal", "opposite angles in a cyclic quadrilateral sum to 180°", "tangent-radius angle is 90°", and so on. These reasons must be stated explicitly — one per step.
Structure every proof the same way: State any angles you can find from the given information (with reasons). Use those to find further angles (with reasons). State the conclusion clearly. If the question says "prove that angle ABC = angle DEF", your final line should literally say "therefore angle ABC = angle DEF" followed by the reason why.
The biggest skill in circle theorem questions is identifying which theorem applies. Here's a quick decision guide based on what you see in the diagram:
Most exam questions use two or three theorems in sequence. Work through the diagram systematically, marking every angle you can find before attempting the one the question asks for. The full specification for circle theorems at each exam board is in the AQA GCSE Maths specification and the Edexcel GCSE Maths specification.