Circle Theorems questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 5 worked examples first, then try the 3 exam-style questions yourself and check your working against the mark scheme.
Circle Properties 2: Tangent Properties: Tangent length and tangent–radius perpendicularity.
Circle Theorems 1: Angles in Circle: Angle at centre = 2 × angle at circumference.
Circle Theorems 2: Tangents & Chords: Alternate-segment theorem, tangent–chord angles.
Circle Theorems 3: Cyclic Quadrilaterals: Opposite angles in cyclic quadrilaterals sum to 180°.
Circle Properties 1: Chords & Intersecting Chords: Chord properties: perpendicular bisector of a chord and intersecting chords.
Circle Theorems worked examples
Worked example 1: 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.
Solution
The tangent is perpendicular to the radius, so OTP is right-angled at T.
\(OP=\sqrt{5^2+12^2}=13\) cm.
Answer: 13 cm
Tip: Look for the right angle between tangent and radius.
Worked example 2: Angle in a semicircle
AB is a diameter and C is on the circle. Angle CAB = \(34^\circ\). Find angle ABC.
Solution
The angle in a semicircle is \(90^\circ\): angle ACB = \(90^\circ\).
Angle ABC \(=180-90-34=56^\circ\).
Answer: 56°
Tip: State the theorem as your reason.
Worked example 3: Alternate segment theorem
The angle between a tangent and a chord AB is \(64^\circ\). C is on the circle in the other segment. Find angle ACB.
Solution
The angle between tangent and chord equals the angle in the alternate segment.
Angle ACB \(=64^\circ\).
Answer: 64°
Tip: Identify the chord first, then the opposite segment.
Worked example 4: Cyclic quadrilateral
In a cyclic quadrilateral, one angle is \(118^\circ\). Find the opposite angle.
Solution
Opposite angles of a cyclic quadrilateral add to \(180^\circ\).
\(180-118=62^\circ\).
Answer: 62°
Tip: This only works if all four vertices lie on the circle.
Worked example 5: Chord length
A chord is 6 cm from the centre of a circle of radius 10 cm. Find the length of the chord.
Solution
The perpendicular from the centre bisects the chord.
Half-chord \(=\sqrt{10^2-6^2}=8\).
Chord \(=16\) cm.
Answer: 16 cm
Tip: Double the half-chord at the end.
Circle Theorems exam-style questions
Try each question before you open the answer. 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 · 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.
Show the answer and mark scheme
M1 4 × 9 = 6 × PD
A1 6 cm
Answer: 6 cm
Question 3 · 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
Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and each answer was re-solved independently before publishing.
Which Edexcel IGCSE Maths papers test circle theorems?
Either Higher tier paper, 1H or 2H, can test circle theorems: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.
Where can I practise more circle theorems questions?
In the IGCSE student hub: Unit 5 (Geometry) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.
IGCSE Math Revision