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Edexcel IGCSE Maths (4MA1) · Unit 5: Geometry

Edexcel IGCSE 4MA1 Circle Theorems: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

The circle theorems link the angles made by chords, radii and tangents. The angle at the centre is twice the angle at the circumference; the angle in a semicircle is $90^{\circ}$ ; angles in the same segment are equal; opposite angles of a cyclic quadrilateral add to $180^{\circ}$ ; a tangent meets a radius at $90^{\circ}$ ; and the angle between a tangent and a chord equals the angle in the alternate segment. Name the theorem as your reason every time.

Unit 5: Geometry
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Notes
Formulas
Worked examples
Mistakes
FAQ

Circle Theorems notes

Angle at the centre and the semicircle

The angle an arc makes at the centre is twice the angle it makes at the circumference. If the angle at the circumference is $38^{\circ}$ , the angle at the centre is $76^{\circ}$ . The angle in a semicircle is the special case: the angle at the centre is $180^{\circ}$ , so the angle at the circumference is $90^{\circ}$ .

Same segment and cyclic quadrilaterals

Angles in the same segment, standing on the same chord, are equal. Opposite angles of a cyclic quadrilateral (all four corners on the circle) add to $180^{\circ}$ : if one is $95^{\circ}$ , the opposite one is $85^{\circ}$ .

Tangents

A tangent is perpendicular to the radius at the point where it touches. Two tangents from the same outside point are equal in length. A tangent of length 15 cm from a point 17 cm from the centre touches a circle of radius $\sqrt{17^{2} - 15^{2}} = 8$ cm.

The alternate segment theorem

The angle between a tangent and a chord equals the angle in the alternate segment, on the other side of the chord. Look for it whenever a tangent and a triangle drawn in the circle meet at the same point.

Chords

The perpendicular from the centre to a chord bisects the chord. When two chords cross at $P$ inside the circle, $A P \times P B = C P \times P D$ : with $A P = 3$ , $P B = 8$ and $C P = 4$ , $P D = 6$ .

Key formulas

Circle theorems: angle at centre = 2 × angle at circumference; angle in a semicircle = 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent ⟂ radius; two tangents from a point are equal; alternate segment theorem; perpendicular from the centre bisects a chord Learn it
Intersecting chords Learn it: $A P \times P B = C P \times P D$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Circle Theorems worked examples

Worked example 1: Angle in a semicircle

AB is a diameter and C is on the circle. Angle CAB = $34^{\circ}$ . Find angle ABC.

Circle with diameter AB and a point C on the circle; angle CAB = 34 degrees

Solution

The angle in a semicircle is $90^{\circ}$ : angle ACB = $90^{\circ}$ .
Angle ABC $= 180 - 90 - 34 = 56^{\circ}$ .

3 steps · 1 to fill in

Tip: State the theorem as your reason.

Worked example 2: Cyclic quadrilateral

In a cyclic quadrilateral, one angle is $118^{\circ}$ . Find the opposite angle.

Cyclic quadrilateral with one angle of 118 degrees; the opposite angle is marked ?

Solution

Opposite angles of a cyclic quadrilateral add to $180^{\circ}$ .
$180 - 118 = 62^{\circ}$ .

3 steps · 1 to fill in

Tip: This only works if all four vertices lie on the circle.

Worked example 3: Tangent and radius

A tangent from P touches a circle of radius 5 cm at T. PT = 12 cm. Find the distance from P to the centre O.

Circle centre O radius 5 cm; the tangent from P touches the circle at T with PT = 12 cm; OP is dashed

Solution

The tangent is perpendicular to the radius, so OTP is right-angled at T.
$O P = \sqrt{5^{2} + 12^{2}} = 13$ cm.

3 steps

Tip: Look for the right angle between tangent and radius.

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Circle Theorems exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 3 marks

A circle has centre O and radius 13 cm. AB is a chord of length 24 cm. Find the shortest distance from O to the chord AB.

Show the answer and mark scheme

M1 perpendicular from O bisects AB: half-chord 12
M1 √(13² − 12²)
A1 5 cm

Answer: 5 cm

Question 2 · easy · 3 marks

A chord of a circle of radius 10 cm is 6 cm from the centre of the circle. Find the length of the chord.

Show the answer and mark scheme

M1 √(10² − 6²) = 8
M1 2 × 8
A1 16 cm

Answer: 16 cm

Question 3 · medium · 2 marks

Two chords AB and CD of a circle intersect at a point P inside the circle. AP = 4 cm, PB = 9 cm and CP = 6 cm. Find PD.

Full worked solution Free, no sign-in

M1 4 × 9 = 6 × PD
A1 6 cm

Answer: 6 cm

Every question here has a full worked solution. See them all →

Question 4 · hard · 3 marks

Chords AB and CD of a circle intersect at P inside the circle. AP = 5 cm, PB = 8 cm and CD = 14 cm. CP is shorter than PD. Find CP.

Show the answer and mark scheme

M1 x(14 − x) = 5 × 8
M1 x² − 14x + 40 = 0 ⇒ (x − 4)(x − 10) = 0
A1 CP = 4 cm

Answer: 4 cm

Question 5 · hard · 3 marks

P is a point outside a circle. A straight line through P meets the circle at A and then at B, so that A lies between P and B, where PA = 4 cm and AB = 12 cm. A second line through P meets the circle at C and D, where PC = 5 cm and C lies between P and D. Find the length of CD.

Show the answer and mark scheme

M1 PA × PB = PC × PD with PB = 16: 4 × 16 = 5 × PD
M1 PD = 12.8
A1 CD = 7.8 cm

Answer: 7.8 cm

Question 6 · hard · 3 marks

PT is a tangent to a circle at T. A line from P meets the circle at A and then at B. PA = 3 cm and AB = 9 cm. Find the length of PT.

Show the answer and mark scheme

M1 PB = 12
M1 PT² = PA × PB = 36
A1 6 cm

Answer: 6 cm

Question 7 · hard · 4 marks

A circle has radius 13 cm. Two parallel chords have lengths 10 cm and 24 cm. The chords are on opposite sides of the centre. Find the distance between the chords.

Show the answer and mark scheme

M1 distance to 10 cm chord: √(13² − 5²)
M1 distance to 24 cm chord: √(13² − 12²)
A1 12 and 5
A1 17 cm

Answer: 17 cm

Question 8 · very hard · 4 marks

AB is a chord of length 16 cm in a circle with centre O. The perpendicular distance from O to AB is 4 cm. (a) Find the radius of the circle, correct to 3 significant figures. (2) (b) Find the size of angle AOB, correct to 3 significant figures. (2)

Show the answer and mark scheme

M1 (a) √(8² + 4²)
A1 8.94 cm (accept 8.94 – 8.95 or √80)
M1 (b) tan(½∠AOB) = 8/4 oe
A1 127° (accept 126.8 – 127)

Answer: (a) 8.94 cm (b) 127°

Common circle theorems mistakes

Halving instead of doubling. The angle at the centre is the bigger one: double the angle at the circumference, or halve the angle at the centre.
A quadrilateral that is not cyclic. Opposite angles add to 180° only when all four corners are on the circle. A quadrilateral with a corner at the centre is not cyclic.
Missing the right angle. Where a tangent meets a radius there is a 90° angle, even if the diagram does not mark it. Look for it before you use Pythagoras or trigonometry.
Shorthand reasons. Write the theorem in words, such as "the angle in a semicircle is $90^{\circ}$ ". "Circle theorem" on its own does not say which one.

Where students lose marks across 4MA1, from the examiner reports →

Circle Theorems FAQ

Are the circle theorems on the 4MA1 formula sheet?

No. None of the circle theorems is printed on the 4MA1 Higher formulae sheet, so learn each one with the words to give as a reason.

How do I spot which circle theorem to use?

Look at what meets where. A diameter means an angle in a semicircle; a tangent means a right angle with the radius or the alternate segment theorem; four points on the circle mean a cyclic quadrilateral; two angles on the same chord mean angles in the same segment.

Can I use a calculator for circle theorems in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Circle Theorems revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 circle theorems questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Circle Theorems: exam questions and answers, notes and worked examples

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Circle Theorems notes

Angle at the centre and the semicircle

Same segment and cyclic quadrilaterals

Tangents

The alternate segment theorem

Chords

Key formulas

Circle Theorems worked examples

Worked example 1: Angle in a semicircle

Solution

Worked example 2: Cyclic quadrilateral

Solution

Worked example 3: Tangent and radius

Solution

Circle Theorems exam-style questions

Question 1 · easy · 3 marks

Question 2 · easy · 3 marks

Question 3 · medium · 2 marks

Full worked solution Free, no sign-in

Question 4 · hard · 3 marks

Question 5 · hard · 3 marks

Question 6 · hard · 3 marks

Question 7 · hard · 4 marks

Question 8 · very hard · 4 marks

Common circle theorems mistakes

Circle Theorems FAQ

Next steps

Practise circle theorems with your progress saved

More Unit 5 topics: Geometry