Circle Theorems

Eight short theorems describe how angles behave in a circle. Learn each one with its exact wording, spot which one a diagram is using, and give it as your reason.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use the angle at the centre, angle in a semicircle and angles in the same segment.
Use opposite angles of a cyclic quadrilateral.
Use tangent–radius, equal tangents and the alternate segment theorem.
State the correct theorem as a reason.

Pause and think

$AB$ is a diameter, so the angle $AOB$ at the centre is $180^\circ$ . If the angle at the centre is always twice the angle at the circumference, what is angle $ACB$ for any point $C$ on the circle?

Section 2 of 14

Core idea

Start hereAny seat on the arc

In a theatre with seats on a circular arc, every seat sees the stage at the same angle. That's a circle theorem in action: angles standing on the same arc are equal. The theorems all follow from one fact: every radius is the same length.

The big idea

Angle at centre = $2\times$ angle at circumference; same segment angles equal; semicircle $90^\circ$ ; cyclic quadrilateral opposite angles $180^\circ$ ; tangent ⟂ radius; tangents from a point equal; alternate segment theorem.

1. Centre and circumference

Angle at the centre is twice the angle at the circumference

position on the arc (°) = 40

Wherever the point moves on the major arc, the angle there stays

60^\circ

: half of

120^\circ

at the centre. (That is also "angles in the same segment are equal".)

Angle at the centre is twice the angle at the circumference

$OA=OP=OB$
Join $P$ (on the circle) through the centre $O$ : two isosceles triangles.
$\angle OPA=\angle OAP=x$
Base angles of isosceles $OAP$ .
$\text{exterior angle at }O=2x$
Exterior angle = sum of opposite interior angles.
$\text{similarly }2y;\ \angle AOB=2(x+y)=2\angle APB$
Add the two parts.

Special case: a diameter subtends

180^\circ

O

, so

90^\circ

at the circumference.

2. More theorems

Angle in a semicircle and a tangent

Any triangle drawn on a diameter has a right angle on the circle. A tangent is perpendicular to the radius at the point of contact.

Angle in a semicircle $=90^\circ$ .
Angles in the same segment are equal.
Cyclic quadrilateral: opposite angles sum to $180^\circ$ .
Tangent ⟂ radius.
Two tangents from a point are equal.
Alternate segment: tangent–chord angle = angle in the opposite segment.

Worked exampleThree quick uses

(a) Centre angle $140^\circ$ : angle at circumference on the same arc? (b) Cyclic quad with an angle $95^\circ$ : opposite angle? (c) Tangent at $T$ , $\angle TOP=50^\circ$ : $\angle OPT$ ?

Goal: Angles.
Strategy: Name the theorem each time.

$140\div2=70^\circ$
Angle at centre is twice.
$180-95=85^\circ$
Cyclic quadrilateral.
$180-90-50=40^\circ$
$\angle OTP=90^\circ$ (tangent ⟂ radius); triangle sum.

Answer

70^\circ

85^\circ

40^\circ

✓ CheckEach answer uses one stated theorem ✓.

Common misconceptionWrong arc The centre angle and circumference angle must stand on the same arc.

Pause and think

$AB$ is a diameter, $C$ on the circle, $\angle CAB=35^\circ$ . Find $\angle CBA$ .

Summary

Learn the theorems and their names.
Look for isosceles triangles with radii.

Cyclic quadrilateral: All four vertices on a circle.
Tangent: Line touching the circle once.

HistoryThales and the semicircle. The angle in a semicircle theorem is named after Thales of Miletus (about

600

BC), often called the first mathematician. Legend says he sacrificed an ox to celebrate proving it. He is one of the first people known to have proved a geometric fact rather than just measuring it.

Fun factFind the centre with a set square. Put the right-angle corner of a set square on the edge of a circular plate and mark where its two edges cross the rim. Those two points are the ends of a diameter (angle in a semicircle). Do it twice and the diameters cross at the centre.

Keep reading for free

Create a free account to read the rest of this lesson: key formulae, worked examples, explore the graph, common mistakes, exam tip, guided practice and more. Your answers are checked and your progress is saved.

Create free account Sign in