G10Geometry & MeasuresHigherGrades 6–9

Circle Theorems

Draw any triangle with its three corners on a circle and the diameter as one side, and the angle opposite the diameter is always $90^{\circ}$ . That is one of a set of circle theorems — reliable angle facts that let you find unknown angles and prove geometric results. AQA expects you to apply them with correct reasons and, at Higher tier, to prove them.

55 min AQA GCSE Maths

The big picture

Most circle theorem questions are angle-chasing: find one angle, use it to find the next, and write a reason for each step. The theorems sit alongside ordinary angle facts — angles in a triangle, isosceles triangles, angles on a straight line.

Two radii always make an isosceles triangle. Spotting those triangles is often the step that unlocks the whole problem, and it is also the key to proving the theorems.

What you'll learn

Recall and apply the circle theorems
Give full reasons using correct mathematical language
Solve multi-step angle problems involving circles
Prove circle theorems using isosceles triangles

Theorems about angles

1. The angle at the centre is twice the angle at the circumference subtended by the same arc.

2. The angle in a semicircle is $90^{\circ}$ .

3. Angles in the same segment are equal.

4. Opposite angles of a cyclic quadrilateral add up to $180^{\circ}$ .

Example 1

A

B

and

C

are points on a circle with centre

O

. Angle

ABC = 38^{\circ}

. Find angle

AOC

, giving a reason.

1Both angles stand on the arc

AC

Answer

AOC = 76^{\circ}

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

Your turnSame method, new numbers — try it before moving on.

An arc subtends an angle of $124^\circ$ at the centre of a circle. What angle does it subtend at the circumference (on the major arc)?

Your answer

Show worked solution

Solution

1The angle at the centre is twice the angle at the circumference.
2 $124 \div 2 = 62^\circ$ .

Answer: $62^\circ$

Example 2

PQRS

is a cyclic quadrilateral with angle

P = 97^{\circ}

. Find angle

R

P

and

R

are opposite angles.

Answer

R = 83^{\circ}

— opposite angles of a cyclic quadrilateral add up to

180^{\circ}

Your turnSame method, new numbers — try it before moving on.

In a cyclic quadrilateral, one angle is $68^\circ$ . Find the opposite angle.

Your answer

Show worked solution

Solution

1Opposite angles of a cyclic quadrilateral add up to $180^\circ$ .
2 $180 - 68 = 112^\circ$ .

Answer: $112^\circ$

Tip — Check that a quadrilateral really is cyclic — all four vertices must lie on the circle, and the centre must not be one of them.

Theorems about tangents and chords

5. A tangent meets a radius at $90^{\circ}$ at the point of contact.

6. Tangents from an external point are equal in length.

7. The perpendicular from the centre to a chord bisects the chord.

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

Example 1A circle has centre

O

and radius 5 cm. Point

P

is 13 cm from

O

. A tangent from

P

touches the circle at

T

. Find

PT

1Angle

OTP = 90^{\circ}

(tangent meets radius at right angles).

PT^{2} = 13^{2} - 5^{2} = 144

Answer

PT = 12

Your turnSame method, new numbers — try it before moving on.

A circle has radius 8 cm. A point $P$ is 17 cm from the centre. Find the length of a tangent from $P$ to the circle, in cm.

Your answer

Show hint

The tangent meets the radius at $90^\circ$ — use Pythagoras.

Show worked solution

Solution

1 $PT^2 = 17^2 - 8^2 = 289 - 64 = 225$ .
2 $PT = 15$ cm.

Answer: $15$ cm

Why this works

A tangent and radius create a right angle, so many circle problems become Pythagoras or trigonometry once that angle is marked.

Writing reasons

Use the standard wording. "Angle at centre is twice angle at circumference" earns the mark; "because of the circle rule" does not.

Give a reason for every angle you find, including ordinary facts like "angles in a triangle add up to $180^{\circ}$ " and "base angles of an isosceles triangle are equal".

Tip — Mark each angle on the diagram as you find it. It keeps the chain of reasoning clear for you and the examiner.

Proving a theorem

To prove the angle in a semicircle is $90^{\circ}$ : let $AB$ be a diameter with centre $O$ and $C$ on the circle.

$OA = OB = OC$ (radii), so triangles $OAC$ and $OBC$ are isosceles. Let angle $OAC = x$ and angle $OBC = y$ ; then angle $OCA = x$ and angle $OCB = y$ .

The angles of triangle $ABC$ add to $180^{\circ}$ : $x + y + (x + y) = 180^{\circ}$ , so $x + y = 90^{\circ}$ , which is angle $ACB$ .

Think like an examiner

Quote theorems in full as reasons.
Identify radii and isosceles triangles.
Use algebra for proofs and state the conclusion.

Remember these

\angle_{\text{centre}} = 2\angle_{\text{circumference}}

Angle at the centre.

\angle A + \angle C = 180^{\circ}

Cyclic quadrilateral.

OT \perp PT

Tangent perpendicular to radius.

Watch out for these

Halving the angle at the circumference to get the centre angle.

The centre angle is double.

Applying the cyclic quadrilateral rule when a vertex is the centre.

All four vertices must be on the circumference.

Writing a reason like "circle theorem".

State the specific theorem.

Using the alternate segment angle on the wrong side of the chord.

Use the segment on the other side of the chord from the tangent angle.

Stretch yourself

Prove that opposite angles of a cyclic quadrilateral add up to $180^{\circ}$ . You may use the fact that the angle at the centre is twice the angle at the circumference.

Hint — Join both remaining vertices to the centre. The two angles at the centre make a full turn.

Reveal solution

SolutionLet

ABCD

be cyclic with centre

O

, and let angle

DAB = x

and angle

BCD = y

. Join

OB

and

OD

. The angle at the centre on the arc

BCD

2x

, and the reflex angle at the centre on the arc

BAD

2y

(angle at centre is twice angle at circumference). These angles make a full turn:

2x + 2y = 360^{\circ}

, so

x + y = 180^{\circ}

\blacksquare

Questions students ask

Do I need to prove all the theorems?

AQA Higher can ask you to prove circle theorems; the semicircle and cyclic quadrilateral proofs are common.

What does "subtended by an arc" mean?

The angle formed by lines drawn from the ends of the arc to a point.

How do I spot the alternate segment theorem?

Look for a tangent and a chord meeting at the same point on the circle, with a triangle drawn inside.

Key takeaways

Angle at centre is twice angle at circumference.
Angle in a semicircle is $90^{\circ}$ ; angles in the same segment are equal.
Opposite angles in a cyclic quadrilateral add to $180^{\circ}$ .
Tangent meets radius at $90^{\circ}$ ; tangents from a point are equal.
Alternate segment theorem links tangent–chord angles to angles in the opposite segment.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 62 marks

$A$ , $B$ and $C$ are points on a circle with centre $O$ . Angle $AOB = 136^\circ$ , and $C$ is on the major arc.

Work out angle $ACB$ . Give a reason for your answer.[2]
Show mark scheme
- B1 $68^\circ$
- B1The angle at the centre is twice the angle at the circumference
Answer: $68^\circ$

Question 2Grade 73 marks

$PT$ is a tangent to a circle with centre $O$ , touching it at $T$ . Angle $TOP = 58^\circ$ .

Work out angle $OPT$ . Give reasons for each stage of your working.[3]
Show mark scheme
- B1Angle $OTP = 90^\circ$ : a tangent is perpendicular to the radius
- M1 $180 - 90 - 58$
- A1 $32^\circ$ with angles in a triangle sum to $180^\circ$
Answer: $32^\circ$

Question 3Grade 83 marks

$ABCD$ is a cyclic quadrilateral. Angle $ABC = 2x + 10$ and angle $ADC = 3x - 5$ .

Find the value of $x$ and the size of angle $ABC$ .[3]
Show mark scheme
- B1Opposite angles of a cyclic quadrilateral sum to $180^\circ$
- M1 $5x + 5 = 180$
- A1 $x = 35$ , angle $ABC = 80^\circ$

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

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