What you'll learn
Circle theorems are a set of rules about angles, tangents and chords in circles. In this guide you will learn the main circle theorems, how to apply them to find missing angles, how to use the tangent and chord properties, and how to set out reasons in a proof-style answer. These are important higher-tier geometry results.
Key terms and definitions
Centre — the middle point of a circle.
Radius — a line from the centre to the circumference.
Chord — a line joining two points on the circumference.
Tangent — a line that touches the circle at one point.
Cyclic quadrilateral — a four-sided shape with all vertices on the circle.
Subtend — an angle "subtended" by an arc is formed by lines from the ends of the arc.
Core concepts
Key angle theorems
The main circle theorems are:
- The angle at the centre is twice the angle at the circumference (subtended by the same arc).
- The angle in a semicircle is 90° (the angle subtended by a diameter).
- Angles in the same segment are equal (subtended by the same arc from the circumference).
- Opposite angles of a cyclic quadrilateral add up to 180°.
Tangent properties
- A tangent meets a radius at 90° (the radius drawn to the point of contact is perpendicular to the tangent).
- Two tangents from the same external point are equal in length, and the line from that point to the centre bisects the angle between them.
- The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment (the inscribed angle subtended by that chord on the other side).
Chord properties
- The perpendicular from the centre to a chord bisects the chord (and vice versa — the line from the centre to the midpoint of a chord is perpendicular to it). This creates right-angled triangles useful with Pythagoras.
Applying the theorems
To find a missing angle, identify which theorem applies from the diagram — look for diameters, tangents, cyclic quadrilaterals or equal arcs. Often you combine several theorems with basic angle facts (angles in a triangle, on a line, etc.). Work step by step.
Giving reasons
In exam answers you must state the reason for each step, quoting the theorem (e.g. "angle at centre is twice angle at circumference", "angle in a semicircle is 90°", "opposite angles of a cyclic quadrilateral sum to 180°"). Marks are awarded for correct reasoning, not just the answer.
Worked examples
Example 1: Angle at the centre
The angle at the centre subtended by an arc is 100°. What is the angle at the circumference?
Half the centre angle: 100 ÷ 2 = 50° (angle at centre is twice angle at circumference).
Example 2: Semicircle
A triangle is drawn in a semicircle with the diameter as one side. What is the angle opposite the diameter?
90° — the angle in a semicircle is a right angle.
Example 3: Cyclic quadrilateral
One angle of a cyclic quadrilateral is 85°. What is the opposite angle?
Opposite angles sum to 180°: 180 − 85 = 95°.
Common mistakes and how to avoid them
Halving/doubling the wrong way. The centre angle is twice the circumference angle subtended by the same arc.
Missing the right angle. A tangent meets the radius at 90°, and the angle in a semicircle is 90°.
Forgetting to give reasons. Quote the theorem for each step.
Misidentifying the alternate segment. The tangent–chord angle equals the angle in the other segment.
Assuming a quadrilateral is cyclic. All four vertices must be on the circle.
Exam technique for Circle Theorems
Learn the theorems and recognise their diagrams.
Look for diameters, tangents and cyclic quadrilaterals as clues.
Combine theorems with basic angle facts.
State a reason for every step, quoting the theorem.
Use chord and Pythagoras when a perpendicular bisects a chord.
Quick revision summary
Circle theorems govern angles, tangents and chords. Key angle results: the angle at the centre is twice the angle at the circumference (same arc); the angle in a semicircle is 90°; angles in the same segment are equal; and opposite angles of a cyclic quadrilateral sum to 180°. Tangent properties: a tangent meets the radius at 90°; two tangents from a point are equal; and the alternate segment theorem says the tangent–chord angle equals the angle in the alternate segment. The perpendicular from the centre bisects a chord, creating right-angled triangles for Pythagoras. To solve problems, identify which theorem applies (look for diameters, tangents, cyclic quadrilaterals), combine theorems with basic angle facts, and work step by step. Crucially, state a reason for every step, quoting the theorem, since marks are for reasoning. The common errors are doubling/halving the wrong way, missing right angles, omitting reasons, and misreading the alternate segment. Learn the theorems, spot the clues, combine them, and justify each step.