Circle Theorems (Years 10–11) — Maths

Angle rules inside circles: centre, semicircle, same segment, cyclic quadrilaterals, tangents.

Look at a Ferris wheel, a clock face, or the spokes fanning out from a bike hub — circles are full of angles that follow strict, reliable rules. Circle theorems are those angle rules. They look intimidating at first, but each one is simply a fact you learn to spot in the diagram, then use to chase a missing angle. Here are the key ones.

Why this matters: circle theorems are a favourite of the higher GCSE, often worth several marks where you must give a reason as well as an answer. More deeply, they train you to justify each step — the beginning of mathematical proof, which matters at A-level and beyond.

An angle drawn on a diameter (in a semicircle) is always 90°

The key theorems

Angle at the centre is twice the angle at the circumference (same arc).

Learn these five as a toolkit. In any problem, scan the diagram and ask "which of these do I recognise here?"

The angle at the centre is twice the angle at the circumference (both standing on the same arc).
The angle in a semicircle is 90° (an angle drawn on a diameter is always a right angle — this is a special case of the first rule).
Angles in the same segment (from the same arc) are equal.
Opposite angles of a cyclic quadrilateral (all four corners on the circle) add to 180°.
A tangent meets the radius at 90°; and two tangents drawn from the same outside point are equal in length.

Angles in the same segment, standing on the same arc, are equal.

Theorem	The rule
Angle at centre	= 2 × angle at circumference (same arc)
Semicircle	Angle on a diameter = 90°
Same segment	Angles from the same arc are equal
Cyclic quadrilateral	Opposite angles add to 180°
Tangent	Meets radius at 90°; two tangents equal

An angle at the centre is 80°. The angle at the circumference from the same arc is half: 80 ÷ 2 = 40°. Working the other way, a circumference angle of 30° gives a centre angle of 30 × 2 = 60°.

A cyclic quadrilateral has one angle of 70°. Its opposite angle is 180 − 70 = 110°, because opposite angles must total 180°.

Opposite angles of a cyclic quadrilateral add to 180°.

Common mistake: the "centre = twice circumference" rule only works when both angles stand on the same arc. Always check which arc each angle sits on before halving or doubling — using the wrong arc is the classic error here.

A tangent meets the radius at exactly 90°.

In a nutshell: centre = 2 × circumference; semicircle = 90°; same segment = equal; cyclic-quad opposite angles = 180°; tangent is perpendicular to the radius.

When the angles are written with a letter

The circle theorems are angle facts, so they become equations in the usual way.

Worked example. In cyclic quadrilateral ABCD, angle A = 3x and the opposite angle C = 2x. Opposite angles of a cyclic quadrilateral add to 180°:

3x + 2x = 180
Collect the terms: 5x = 180
Divide by 5: x = 36

Check: A = 108°, C = 72°, and 108 + 72 = 180. ✓