Circle Theorems — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the angle relationships inside and around a circle. By the end of this guide you should be able to state each of the circle theorems and use them to find missing angles.

You should also be able to give the correct reason for every step in the standard wording examiners require, combine several theorems in one problem, and use the tangent and chord properties alongside the angle theorems.

The organising idea is that each theorem is triggered by a specific arrangement, so the skill is recognising the arrangement rather than recalling a list. A right angle inside a triangle only appears when one side is a diameter. Two angles are equal only when they stand on the same arc from the same side. Two angles add to 180° only when the four points lie on the circle. So the working method is to hunt the diagram for these configurations — find the diameter, find the two angles on the same arc, find the four points on the circumference — and the theorem announces itself. Marks here are split evenly between the angle and the reason, so naming what you spotted is half the answer.

Key terms and definitions

Radius — a line from the centre to the circumference.

Chord — a straight line joining two points on the circumference.

Diameter — a chord through the centre.

Tangent — a line touching the circle at exactly one point.

Arc — part of the circumference.

Segment — the region between a chord and its arc.

Subtend — an angle subtended by an arc is formed by lines drawn from the ends of that arc.

Cyclic quadrilateral — a four-sided shape with all four vertices on the circumference.

Core concepts

The angle at the centre

The angle at the centre is twice the angle at the circumference, when both stand on the same arc.

So if an angle at the circumference is 35°, the angle at the centre subtended by the same arc is 70°.

The arrangement to spot is two angles standing on the same two points, one with its vertex at the centre and one on the circumference.

Reason to write: "the angle at the centre is twice the angle at the circumference".

The angle in a semicircle

The angle in a semicircle is 90°.

Whenever a triangle has a diameter as one of its sides and its third vertex on the circumference, the angle at that vertex is a right angle.

This is really the previous theorem in a special case: the angle at the centre along a diameter is a straight 180°, so the angle at the circumference is half of that.

Finding the diameter is the trigger. A line passing through the centre is a diameter even if the question does not label it as one.

Reason to write: "the angle in a semicircle is 90°".

Angles in the same segment

Angles in the same segment are equal — two angles standing on the same arc, with both vertices on the same side of it, are the same size.

The arrangement looks like two triangles sharing a base, with their apexes both on the circumference.

The phrase "same segment" matters. Angles on opposite sides of the chord are not equal; they are supplementary, which is the cyclic quadrilateral result below.

Reason to write: "angles in the same segment are equal".

Cyclic quadrilaterals

Opposite angles of a cyclic quadrilateral add to 180°.

The trigger is four points all lying on the circumference, joined to make a quadrilateral.

So if one angle is 115°, the angle opposite it is 65°. The other pair also totals 180°, independently.

A consequence sometimes tested: the exterior angle of a cyclic quadrilateral equals the interior opposite angle, which follows from the angles on a straight line.

Reason to write: "opposite angles of a cyclic quadrilateral add up to 180°".

Tangent and radius

A tangent meets a radius at 90° at the point of contact.

This is the most frequently used theorem of all, because it creates right-angled triangles that Pythagoras and trigonometry can then be applied to.

Whenever a tangent appears, draw in the radius to the point of contact if it is not already there.

Reason to write: "the angle between a tangent and a radius is 90°".

Two tangents from a point

Two tangents drawn from the same external point are equal in length.

That makes the triangle formed by the two tangents and the chord joining their contact points isosceles, which brings the equal base angles into play.

The quadrilateral formed by the two radii, the two tangents and the centre also has two right angles, so its other two angles must total 180°.

Reason to write: "tangents from an external point are equal".

Perpendicular from the centre to a chord

A perpendicular from the centre to a chord bisects the chord.

Equivalently, the line from the centre to the midpoint of a chord is perpendicular to it.

This creates a right-angled triangle with the radius as hypotenuse, half the chord as one side, and the distance from the centre as the other — which is how chord length questions are solved.

Reason to write: "the perpendicular from the centre to a chord bisects it".

The alternate segment theorem

The angle between a tangent and a chord equals the angle in the alternate segment.

The alternate segment is the one on the other side of the chord from the angle you are looking at.

This is the hardest theorem to spot, and the arrangement is specific: a tangent, a chord drawn from the point of contact, and an angle in the segment beyond that chord.

Reason to write: "the alternate segment theorem".

Working through a multi-step problem

Most exam questions chain two or three theorems together, and the intermediate angles are not marked.

Work forwards from what you know, marking each angle on the diagram as you find it and writing its reason beside it.

Look first for the arrangements that are easiest to spot: a diameter, a tangent, or four points on the circumference. Each of those immediately gives an angle.

Remember the ordinary angle facts remain available — angles in a triangle total 180°, angles on a straight line total 180°, angles at a point total 360°, and any triangle with two radii as sides is isosceles, since all radii are equal.

That last point is easy to overlook and frequently supplies the missing step.

Giving reasons

Reason marks are awarded for the standard phrases, and loose paraphrases often score nothing.

Write "angles in the same segment are equal", not "they're on the same arc so they match". Write "the angle in a semicircle is 90°", not "it's a right angle because of the diameter".

Where a step uses an ordinary angle fact rather than a circle theorem, name that too — "base angles of an isosceles triangle are equal, since both are radii".

Worked examples

Example 1: The angle at the centre

Points A and B lie on a circle with centre O. Angle ACB at the circumference is 42°, where C is on the major arc. Find angle AOB.

The angle at the centre and the angle at the circumference both stand on the arc AB.

The angle at the centre is twice the angle at the circumference.

So angle AOB = 2 × 42 = 84°, with the reason "the angle at the centre is twice the angle at the circumference".

Example 2: A diameter and an isosceles triangle

AB is a diameter of a circle with centre O, and C is a point on the circumference. Angle CAB is 34°. Find angle ACB and angle ABC.

AB is a diameter, so angle ACB is the angle in a semicircle: 90°.

Reason: "the angle in a semicircle is 90°".

The angles of triangle ABC total 180°, so angle ABC = 180 − 90 − 34 = 56°.

Reason: "angles in a triangle add up to 180°".

Both reasons are needed, and the diameter is what unlocked the first one.

Example 3: A tangent problem

A tangent touches a circle of radius 5 cm at point T. A point P outside the circle is 13 cm from the centre O. Find the length PT.

The tangent meets the radius at 90°, so triangle OTP has a right angle at T.

Reason: "the angle between a tangent and a radius is 90°".

OP is the hypotenuse at 13 cm and OT is the radius at 5 cm, so by Pythagoras PT² = 169 − 25 = 144.

So PT = 12 cm.

Drawing in the radius to the point of contact was the step that turned a circle problem into an ordinary right-angled triangle.

Common mistakes and how to avoid them

Giving an angle with no reason. Half the marks are for the reasons, in the standard wording.

Using "angles in the same segment" for angles on opposite sides of the chord. Those are supplementary, not equal.

Missing the isosceles triangle formed by two radii. All radii are equal, so those base angles match.

Failing to draw in the radius at a tangent. It creates the right angle the question depends on.

Halving instead of doubling at the centre. The centre angle is the larger one.

Assuming a line through the middle of the diagram is a diameter. It must pass through the centre.

Paraphrasing the theorem names. Use the standard phrases.

Exam technique for "Circle Theorems"

Mark every angle you find on the diagram as you go, with its reason beside it. The next step usually becomes visible once two or three are labelled.

Hunt for the trigger arrangements first: a diameter gives a right angle, a tangent gives a right angle with the radius, four points on the circumference give supplementary opposite angles.

Draw in the radius to any point of contact, and to any point where it might create an isosceles triangle.

Write reasons in the standard wording, and name ordinary angle facts as well as circle theorems.

Check the angles in any triangle you use total 180°, as a running check on the whole chain.

If a question says "give reasons for your answer", expect a reason mark for each step, not one at the end.

Quick revision summary

Each theorem is triggered by an arrangement — find the arrangement and the theorem follows.

Angle at the centre is twice the angle at the circumference on the same arc.

Angle in a semicircle is 90° — the trigger is a diameter as one side of the triangle.

Angles in the same segment are equal — same arc, same side of the chord.

Opposite angles of a cyclic quadrilateral add to 180° — the trigger is four points on the circumference.

Tangent meets radius at 90°, and two tangents from a point are equal, making an isosceles triangle.

A perpendicular from the centre bisects a chord, creating a right-angled triangle with the radius as hypotenuse.

Alternate segment theorem: the angle between a tangent and a chord equals the angle in the segment on the other side.

Two radii always make an isosceles triangle, and the ordinary angle facts stay available throughout. Give a named reason for every step.