Which Circle Theorem Should I Use? A Visual Guide

Remembering the circle theorems is often easier than recognising which one a question needs. Several overlapping triangles and chords can make a diagram difficult to read. Start with the information you have been given: a diameter gives you a right angle, while a tangent becomes useful once you draw the radius to its point of contact.

It also helps to trace the two arms of the angle you are trying to find. Where they meet the circle tells you which chord or arc the angle stands on.

How to choose a circle theorem

If you need to find an angle, the features below can help you decide where to start. For a length, see the length relationships below. Some questions need more than one theorem.

Centre marked?Angles at the centre and circumferenceBoth angles must stand on the same arc.Two angles facing one chord?Angles in the same segmentThe vertices must be on the same side of the chord.A confirmed diameter?An angle in a semicircleThe angle opposite the diameter is a right angle.Four points on the circle?Opposite angles in a cyclic quadrilateralOpposite angles add to 180°.Tangent and radius meet?A radius and a tangentThe radius meets the tangent at 90° at the point of contact.Tangent meets a chord?The alternate-segment theoremCompare the tangent–chord angle with the angle across the chord.

These questions also use ordinary geometry: angles on a straight line total $180^{\circ}$ , angles in a triangle total $180^{\circ}$ , and equal radii make isosceles triangles.

Circle theorems for finding angles

A chord joins two points on the circle. An arc is part of the circumference between two points. A tangent touches the circle at one point. The relevant arcs, chords and angles are marked in gold in the diagrams.

Look for the same arc

Angles at the centre and circumference

The central angle AOB is 128 degrees and the circumference angle ACB is 64 degrees. Both intercept the minor arc AB, which does not contain C.

The angle at the centre is twice the angle at the circumference standing on the same arc.

∠ A O B = 2 ∠ A C B

Trace the arms of both angles to $A$ and $B$ . In this diagram, $C$ lies on the major arc, so both marked angles stand on the minor arc $A B$ .

The reflex-angle case

Use the arc that does not contain the circumference vertex. If that vertex lies on the minor arc instead, the matching central angle is reflex. Check which arc each angle stands on before applying the theorem.

Same chord + same segment

Angles in the same segment

Points P and Q lie on the same side of highlighted chord AB. Angles APB and AQB are both 50 degrees.

Angles standing on the same chord, with their vertices in the same segment, are equal.

∠ A P B = ∠ A Q B

Both pairs of arms end at $A$ and $B$ ; $P$ and $Q$ are on the same side of chord $A B$ . You can move the vertex along that arc without changing the angle.

Opposite sides of the chord

Vertices on opposite sides of the chord do not generally give equal angles. Those two angles add to $180^{\circ}$ .

Check that the line passes through the centre

The angle in a semicircle

AB is a diameter through centre O. C is on the circle, and angle ACB is marked as a right angle.

An angle subtended by a diameter at another point on the circle is a right angle.

∠ A C B = 90^{\circ}

The right angle is at $C$ , opposite diameter $A B$ —not at one of the diameter’s endpoints. You can then use the triangle angle sum or Pythagoras.

Is it a diameter?

A long chord is not automatically a diameter. Look for the centre on it, or a statement that it is a diameter.

All four vertices must lie on the circle

Cyclic quadrilaterals

A, B, C and D lie in order on a circle. The opposite interior angles at B and D are labelled alpha and beta.

A quadrilateral whose four vertices lie on one circle is cyclic. Its opposite interior angles are supplementary.

α + β = 180^{\circ}

Pair opposite corners, not adjacent ones. An exterior angle of a cyclic quadrilateral also equals the opposite interior angle.

All four vertices

Make sure all four vertices are stated to be on the circle. Otherwise, you need to prove that the quadrilateral is cyclic before using this theorem.

Practise cyclic quadrilaterals →

Join the centre to the point of contact

A radius and a tangent

OT is a radius and a vertical tangent touches the circle at T. A square marks the 90-degree angle between them.

A tangent is perpendicular to the radius drawn to its point of contact.

O T ⟂ tangent at T

If the radius is missing, drawing it can create a right-angled triangle. Another radius may then give you an isosceles triangle as well.

The point of contact

The radius must meet the tangent at the contact point. A chord reaching that point is not generally perpendicular to the tangent.

Angle between a tangent and a chord

The alternate-segment theorem

A tangent at A meets chord AB. The marked tangent–chord angle equals angle ACB on the opposite side of chord AB; both are theta.

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

Start at the point of contact $A$ , follow chord $A B$ , then find the circumference angle facing that chord in the opposite segment. The two marked angles are equal.

Choose the correct side

The two tangent rays give supplementary tangent–chord angles. Match the marked angle to the alternate segment; do not simply choose whichever angle looks acute.

Explore chords and tangents →

Worked examples

In a written solution, state the theorem that justifies each equation. This helps someone reading your working follow the steps, especially when you use more than one theorem.

EXAMPLE 1 · CENTRE AND CIRCUMFERENCE

Finding an angle at the circumference

In the centre diagram above, $∠ A O B = 128^{\circ}$ . Point $C$ is on the major arc $A B$ . Find $∠ A C B$ .

Identify the matching arc. The angle at $C$ stands on the minor arc $A B$ . The marked central angle stands on that arc too.
Use the centre–circumference relationship. $∠ A C B = \frac{128^{\circ}}{2} = 64^{\circ}$

A vertex on the minor arc. If a point $D$ were on the minor arc $A B$ , its angle would face the major arc. Then $∠ A D B = \frac{360^{\circ} - 128^{\circ}}{2} = 116^{\circ}$ . The location of the vertex changes the answer.

EXAMPLE 2 · CYCLIC QUADRILATERALS

Finding unknown angles in a cyclic quadrilateral

$A B C D$ is cyclic. Its opposite angles are ${(3 x + 20)}^{\circ}$ and ${(2 x + 10)}^{\circ}$ . Find $x$ and both angles.

Cyclic quadrilateral ABCD with opposite angles at B and D labelled (3x+20) degrees and (2x+10) degrees.

Write the angle equation. Opposite angles in a cyclic quadrilateral add to $180^{\circ}$ , so their degree measures satisfy $\begin{array}{c} (3 x + 20) + (2 x + 10) \\ = 180 \end{array}$
Solve the equation.
$5 x + 30 = 180$ $5 x = 150 \Rightarrow x = 30$
Calculate both angles. $\begin{aligned} ∠ A B C & = {(3 \times 30 + 20)}^{\circ} \\ = 110^{\circ} \end{aligned}$ $\begin{aligned} ∠ A D C & = {(2 \times 30 + 10)}^{\circ} \\ = 70^{\circ} \end{aligned}$
The angles total $180^{\circ}$ . Notice that $x = 30$ is not itself either of the marked angles.

EXAMPLE 3 · TANGENTS AND RADII

Using a tangent and equal radii

$O$ is the centre. The line through $T$ is tangent, and the marked tangent–chord angle is $38^{\circ}$ . Find $x = ∠ O T B$ and $y = ∠ T O B$ .

A tangent at T makes 38 degrees with chord TB. Radii OT and OB form triangle OTB; x is its angle at T and y is its angle at O.

Find the right angle. Radius $O T$ is perpendicular to the tangent. The marked angles split that right angle: $x = 90^{\circ} - 38^{\circ} = 52^{\circ}$
Use equal radii. Since $O T = O B$ , triangle $O T B$ is isosceles. Its angle at $B$ is also $52^{\circ}$ .
Find the angle at the centre. $y = 180^{\circ} - 2 (52^{\circ}) = 76^{\circ}$

Another possible approach. The alternate-segment theorem would give an equal angle elsewhere on the circumference. That angle is not needed here: radius–tangent perpendicularity and equal radii are enough.

Circle theorems for lengths

Several circle theorems relate lengths rather than angles. Check for equal radii or equal tangent segments from one external point. If lines pass through the circle, one of the product relationships below may apply. These relationships can be proved using similar triangles.

Two tangents from one point

If $P A$ and $P B$ are tangent segments from the same external point $P$ , then $P A = P B$ .

Chords intersect inside

For chords $A B$ and $C D$ meeting at $P$ inside the circle:

P A \times P B = P C \times P D

Two secants from outside

From the same external point $P$ , with near points $A, C$ and far points $B, D$ on the two secants:

P A \times P B = P C \times P D

Here each product is external part × whole secant, not external × internal.

A tangent and a secant

If $P T$ is tangent and the secant meets the circle at near point $A$ and far point $B$ :

P T^{2} = P A \times P B

For example, an external part of $3$ and an internal part of $5$ make the whole secant $8$ . Use $P T^{2} = 3 \times 8 = 24$ , so the positive tangent length is $P T = 2 \sqrt{6}$ in the same length units.

See the configurations and practise length relationships →

Three questions to try

For each question, decide which theorem applies and calculate the missing angle. The solutions are below each question.

01 / SAME CHORD

Same side or opposite side?

$A, B, P, Q$ lie on one circle. $P$ and $Q$ are on the same side of chord $A B$ . If $∠ A P B = 47^{\circ}$ , find $∠ A Q B$ .

Show solution

$47^{\circ}$ . Both angles stand on chord $A B$ in the same segment. There is no central angle here to double.

02 / DIAMETER

Which corner is the right angle?

$A, B, C$ lie on a circle. $A B$ is a diameter and $∠ B A C = 32^{\circ}$ . Find $∠ A B C$ .

Show solution

The diameter gives $∠ A C B = 90^{\circ}$ . Then $∠ A B C = 180^{\circ} - 90^{\circ} - 32^{\circ} = 58^{\circ}$ .

03 / TANGENT

Which angles split the right angle?

A tangent touches a circle at $A$ . Radius $O A$ and chord $A B$ make $28^{\circ}$ . Find the acute tangent–chord angle.

Show solution

The radius is perpendicular to the tangent. The acute tangent–chord angle is $90^{\circ} - 28^{\circ} = 62^{\circ}$ . The angle using the other tangent ray is $118^{\circ}$ .

A note on diagrams

Circle diagrams are not always drawn to scale. A chord may look like a diameter without passing through the centre, and two angles that look equal may have different values. Use the marked information and the conditions of the theorem rather than measuring the picture.

Which circle theorem
should I use?

How to choose a circle theorem

Circle theorems for finding angles

Angles at the centre and circumference

Angles in the same segment

The angle in a semicircle

Cyclic quadrilaterals

A radius and a tangent

The alternate-segment theorem

Worked examples

Finding an angle at the circumference

Finding unknown angles in a cyclic quadrilateral

Using a tangent and equal radii

Circle theorems for lengths

Three questions to try

Same side or opposite side?

Which corner is the right angle?

Which angles split the right angle?

A note on diagrams

Further reading