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Cambridge IGCSE Maths (0580) · Topic 4: Geometry · 4.7, 4.8

Cambridge IGCSE Maths 0580 Circle Theorems Questions

Circle Theorems questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.

Practise circle theorems →Circle Theorems notes

What the syllabus asks

Summarised in our own words: check the exact wording in the official 0580 syllabus.

Circle Theorems questions: Core

For everyone: Core students (Papers 1 and 3) and Extended students (Papers 2 and 4).

Question 1 Core non-calculator · easy

\(AB\) is a diameter of a circle and \(C\) is a point on the circle. Angle \(ABC = 48°\).
Find angle \(BAC\).

Show the worked answer
  1. The angle in a semicircle is \(90°\), so angle \(ACB = 90°\).
  2. Angle \(BAC = 180° - 90° - 48° = 42°\).

Answer: \(42°\)

Question 2 Core non-calculator · easy

\(PT\) is a tangent to a circle with centre \(O\), touching it at \(T\). Angle \(TOP = 62°\).
Find angle \(OPT\).

Show the worked answer
  1. A tangent meets the radius at \(90°\), so angle \(OTP = 90°\).
  2. Angle \(OPT = 180° - 90° - 62° = 28°\).

Answer: \(28°\)

Question 3 Core calculator · easy · 2 marks

\(AB\) is a diameter of a circle, and \(C\) is a point on the circle. \(AB = 13\) cm and \(AC = 5\) cm. Find \(BC\).

Show the answer and mark scheme
  • M1 angle ACB = 90° and 13² − 5²
  • A1 12 cm

Answer: \(12\) cm

Question 4 Core non-calculator · medium · 2 marks

\(AB\) is a diameter of a circle and \(C\) is on the circle. Angle \(CAB = x^\circ\) and angle \(CBA = 2x^\circ\). Find \(x\).

Show the answer and mark scheme
  • M1 x + 2x = 90
  • A1 30

Answer: \(x = 30\)

Question 5 Core calculator · medium · 2 marks

A tangent from a point \(P\) touches a circle with centre \(O\) at \(T\). \(OT = 6\) cm and \(OP = 10\) cm. Find the length \(PT\).

Show the answer and mark scheme
  • M1 angle OTP = 90° and 10² − 6²
  • A1 8 cm

Answer: \(8\) cm

Circle Theorems questions: Extended

Extended content, for Papers 2 and 4. Core students can skip these.

Question 6 Extended non-calculator · medium

\(A\), \(B\) and \(C\) are points on a circle with centre \(O\). Angle \(AOB = 130°\) and \(C\) is on the major arc.
Find angle \(ACB\).

Show the worked answer
  1. The angle at the centre is twice the angle at the circumference: \(130 \div 2 = 65°\).

Answer: \(65°\)

Question 7 Extended non-calculator · medium

\(ABCD\) is a cyclic quadrilateral. Angle \(A = 78°\) and angle \(B = 95°\).
Find angles \(C\) and \(D\).

Show the worked answer
  1. Opposite angles of a cyclic quadrilateral add to \(180°\).
  2. \(C = 180 - 78 = 102°\) and \(D = 180 - 95 = 85°\).

Answer: \(C = 102°\), \(D = 85°\)

Question 8 Extended non-calculator · hard

A tangent touches a circle at \(A\). \(AB\) is a chord, and the angle between the tangent and \(AB\) is \(58°\). \(C\) is a point on the circle in the alternate segment.
Find angle \(ACB\).

Show the worked answer
  1. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Answer: \(58°\)

Question 9 Extended calculator · hard

Two tangents from a point \(P\) touch a circle with centre \(O\) and radius \(5\) cm at \(A\) and \(B\). \(PA = 12\) cm.
Find (a) \(PB\) (b) \(OP\).

Show the worked answer
  1. (a) Tangents from an external point are equal: \(PB = 12\) cm.
  2. (b) Angle \(OAP = 90°\), so \(OP = \sqrt{5^2 + 12^2} = 13\) cm.

Answer: (a) \(12\) cm (b) \(13\) cm

Question 10 Extended calculator · hard

A chord of length \(16\) cm is drawn in a circle of radius \(10\) cm.
How far is the chord from the centre of the circle?

Show the worked answer
  1. The perpendicular from the centre bisects the chord: half the chord is \(8\) cm.
  2. Distance \(= \sqrt{10^2 - 8^2} = \sqrt{36} = 6\) cm.

Answer: \(6\) cm

Examiner Insights: common mistakes in IGCSE Maths 0580, with fixes (our analysis of public sources) →

Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).

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Questions students ask

Which Cambridge 0580 papers can test circle theorems?

Circle Theorems is on both tiers (syllabus 4.7, 4.8), so it can come up on Core Papers 1 (non-calculator) and 3 (calculator), and Extended Papers 2 (non-calculator) and 4 (calculator). Extended candidates also need the Extended parts listed on this page. Half of each tier's marks are on a non-calculator paper, so practise the non-calculator questions without one.

Are these Cambridge past-paper questions?

No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.

Where can I practise more circle theorems questions?

In the 0580 practice for topic 4 (Geometry), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.

More geometry questions