Mathematics 0580 · syllabus C4.7, E4.7, E4.8
Cambridge IGCSE Mathematics 0580 — Circle theorems topical past papers
Angle facts in circles, from the semicircle and tangent theorems to cyclic quadrilaterals and the alternate segment theorem, with the reasons to give.
- Questions
- 214
- Years
- 2009–2026
- With mark scheme
- 90%
- Questions 2022–2026
- 67
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Circle theorems covers
Circle theorems are angle facts that hold in every circle. Core candidates use two of them: the angle in a semicircle is 90°, so a triangle drawn on a diameter has a right angle at the circumference, and a tangent meets the radius at its point of contact at 90°. You need the vocabulary of a circle as well: centre, radius, diameter, chord, tangent, arc, sector and segment.
Extended candidates use four more angle theorems: the angle at the centre is twice the angle at the circumference standing on the same arc, angles in the same segment are equal, opposite angles of a cyclic quadrilateral add up to 180°, and the angle between a tangent and a chord equals the angle in the alternate segment. They also use three symmetry properties: equal chords are the same distance from the centre, the perpendicular bisector of a chord passes through the centre, and the two tangents from a point outside the circle are equal in length.
Every answer needs reasons, written as the properties themselves, such as 'angle in a semicircle' or 'opposite angles of a cyclic quadrilateral'. The two Core theorems can be set on all four papers; the rest are Extended, examined only on Papers 2 and 4. Circle theorem questions also use the angle facts of triangles, parallel lines and polygons, and a right angle from a theorem often opens the way to Pythagoras' theorem or trigonometry.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C4.7 Circle theorems
- E4.7 Circle theorems I
- E4.8 Circle theorems II
What you should be able to do
- Use the angle in a semicircle and the right angle between a tangent and a radius
- Use the angle at the centre, angles in the same segment and cyclic quadrilaterals (Extended)
- Use the alternate segment theorem (Extended)
- Use the symmetry properties of chords and tangents (Extended)
- Give a correct reason for every angle found
- Combine circle theorems with the angle facts of triangles, polygons and parallel lines
Key ideas
- Angle in a semicircle = 90°; angle between a tangent and a radius = 90°
- Angle at the centre = 2 × angle at the circumference on the same arc (Extended)
- Angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add up to 180° (Extended)
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment (Extended)
- Tangents from an outside point are equal, and the perpendicular bisector of a chord passes through the centre (Extended)
How Circle theorems has been examined
From 2022 to 2026, Circle theorems was on Paper 3 · Calculator (Core) in 10 of 14 exam series, worth about 4% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 11 | 2% | 8 of 14 |
| Paper 2 · Non-calculator (Extended) | 24 | 4% | 12 of 14 |
| Paper 3 · Calculator (Core) | 17 | 4% | 10 of 14 |
| Paper 4 · Calculator (Extended) | 15 | 3% | 11 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 4.
Filed in Momentum under: Circle Theorem.
A worked example
A, B, C and D lie on a circle with centre O, with B on the major arc AC and D on the minor arc AC. Angle AOC = 136°. Find angle ABC and angle ADC, giving reasons.
- Angle ABC stands on the minor arc AC, the same arc as angle AOC at the centre.
- The angle at the centre is twice the angle at the circumference, so angle ABC = 136° ÷ 2 = 68°.
- ABCD is a cyclic quadrilateral, and B and D are opposite vertices.
- Opposite angles of a cyclic quadrilateral add up to 180°, so angle ADC = 180° − 68° = 112°.
- Check: angle ADC stands on the major arc, whose angle at the centre is the reflex angle 360° − 136° = 224°, and half of 224° is 112°.
Answer: Angle ABC = 68°; angle ADC = 112°
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Using the angle at the centre theorem when the two angles stand on different arcs: both must be subtended by the same arc.
- Treating a quadrilateral as cyclic when one of its vertices is the centre: all four vertices must lie on the circumference.
- Writing a vague reason such as 'circle rule': the property must be stated in words to earn the reason mark.
- Missing the isosceles triangle formed by two radii: the radii are equal, so the base angles are equal, and this is often the first step.
- Applying the alternate segment theorem to the wrong angle: the angle between the tangent and the chord equals the angle in the segment on the other side of that chord.
Exam technique
- Mark equal radii and the right angles at tangents on the diagram first; they unlock most questions.
- Write each angle with its three-letter name, its value and its reason, one per line.
- When several points lie on the circle, look for an angle at the centre or a cyclic quadrilateral.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.