Cambridge IGCSE Mathematics 0580

Circle theorems

The circle theorems are eight facts about angles in circles that Cambridge IGCSE 0580 (Extended) expects you to know, use — and, the part most students lose marks on, name as a reason. In a recent Cambridge examiner report, giving correct geometrical reasons was called the most challenging question on the entire paper, with a significant share of candidates not attempting it.

This guide states each theorem the way the mark scheme accepts it, works through an angle chase with full reasons, and lists the wordings that do and don’t score.

Updated 20 August 2026

The eight circle theorems

1. Angle at the centre

The angle subtended by an arc at the centre is twice the angle subtended at the circumference by the same arc: $\angle AOB = 2\angle ACB$ . This is the parent theorem — two others below are special cases of it.

2. Angle in a semicircle

The angle in a semicircle is 90°. If $AB$ is a diameter, any point $C$ on the circle gives $\angle ACB = 90^\circ$ . (It’s the angle-at-centre theorem with the centre angle at 180°.)

3. Angles in the same segment

Angles subtended by the same arc, in the same segment, are equal. Two angles “standing on” the same chord from the same side match exactly.

4. Opposite angles of a cyclic quadrilateral

A quadrilateral with all four vertices on a circle has opposite angles summing to 180°: $\angle A + \angle C = 180^\circ$ .

5. Tangent perpendicular to radius

A tangent meets the radius drawn to the point of contact at 90°.

6. Tangents from an external point

The two tangents drawn from the same external point are equal in length — which also makes the triangle they form with the chord of contact isosceles.

7. Alternate segment theorem

The angle between a tangent and a chord equals the angle in the alternate segment — the angle the chord subtends on the far side of the circle. This is the theorem students find hardest to spot; look for a tangent plus a triangle whose base is a chord through the contact point.

8. Perpendicular from the centre to a chord

The perpendicular from the centre to a chord bisects the chord. Often combined with Pythagoras to find lengths rather than angles.

Reasons that score the mark

When a question says “give a geometrical reason”, the mark is for naming the theorem in recognisable words — not for restating the numbers. Wordings the mark schemes accept:

“Angle at the centre is twice the angle at the circumference”
“Angle in a semicircle is 90°”
“Angles in the same segment are equal”
“Opposite angles of a cyclic quadrilateral add up to 180°”
“Tangent is perpendicular to the radius”
“Tangents from an external point are equal”
“Alternate segment theorem”

What doesn’t score: vague gestures like “because of the circle theorem”, “angles in a circle”, or restating the calculation (“because 104 ÷ 2 = 52”). If your reason could apply to any diagram whatsoever, it isn’t a reason.

Worked example: an angle chase with reasons

Worked example

A, B and C lie on a circle with centre O. Angle AOC = 116° (the angle at the centre, on the same arc AC as B). T is a tangent at C, and the angle between the tangent and chord CB is 44°. Find angle ABC and angle BCA, giving reasons.

Step 1 — angle at the circumference. $\angle ABC = 116^\circ \div 2 = 58^\circ$ , because the angle at the centre is twice the angle at the circumference on the same arc.

Step 2 — alternate segment. The 44° angle between tangent and chord $CB$ equals the angle in the alternate segment, so $\angle BAC = 44^\circ$ .

Step 3 — angles in a triangle. $\angle BCA = 180^\circ - 58^\circ - 44^\circ = 78^\circ$ .

Every step is one theorem plus one line of arithmetic — and each reason is named, not implied. That’s the structure the mark scheme is built around: typically one mark for each correct angle and one for each correct reason.

How 0580 examines circle theorems

Circle theorems are Extended-tier content, appearing on Paper 2 (non-calculator) and Paper 4 (calculator), usually as one structured question worth 2–6 marks. The standard format gives a marked diagram (always not to scale — measuring it proves nothing) with one or two known angles, then asks for unknown angles with or without reasons. Harder variants chain three or more theorems, often hiding an isosceles triangle made of two radii along the way — if two sides of a triangle are radii, its base angles are equal, and examiners use this constantly.

The mistakes that lose the marks

1.Calculating correctly but giving no reason (or a vague one)
The reason is a separate mark. Train the habit: every angle you write down gets a named theorem next to it, even in practice. “Angles in a circle” scores nothing.
2.Halving when you should double (and vice versa)
The angle at the centre is the big one: $\angle AOB = 2\angle ACB$ . Check which point your angle sits at before touching the 2.
3.Missing the radii isosceles triangle
Any triangle with two vertices on the circle and one at the centre has two sides that are radii — equal — so its base angles are equal. Mark every radius in the diagram before starting.
4.Applying the alternate segment theorem to the wrong angle
The tangent-chord angle equals the angle in the alternate segment — the one on the other side of the chord. Trace the chord from the tangent point and jump to the angle standing on it from the far side.
5.Assuming a quadrilateral is cyclic when a vertex is the centre
Opposite angles sum to 180° only when all four vertices lie on the circle. If one vertex is $O$ , it isn’t a cyclic quadrilateral — use the angle-at-centre theorem instead.

Common questions

What are the 9 circle theorems?

Different sources count them differently — some split “perpendicular from the centre bisects a chord” into two directions, giving nine. For 0580 you need the eight facts above; how you number them doesn’t matter, but the wording of each reason does.

Do you always have to give reasons?

Only when the question asks — but it usually does, and the reason typically carries its own mark. If the command is “find angle x” alone, a correct value scores; if it says “give a reason for your answer”, the value without the theorem name loses that mark.

Are circle theorems on the Core papers?

Core covers basic circle vocabulary and the semicircle and tangent-radius facts; the full theorem set with reasoned angle chases is Extended (Papers 2 and 4).

Practise circle theorems against real mark schemes

Quanta has real Cambridge IGCSE Maths 0580 past-paper questions, broken into skill checkpoints, marked criterion by criterion the way examiners mark — and it tracks which skills you’re missing. Free for individual students.

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