Angle at the Centre

The angle at the centre is twice the angle at the circumference: double it or halve it, find a reflex angle at the centre, and write the reason in the exact words.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 10 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

Centre and Circumference

Pick two points A and B on a circle. Join them to the centre O: that makes the angle at the centre.

Join A and B to another point C on the circle: that makes the angle at the circumference.

Rule

The angle at the centre is twice the angle at the circumference. Both angles must stand on the same arc AB.

Key idea

Write the reason in these exact words: "angle at the centre is twice the angle at the circumference". The mark scheme looks for them. Saying it the other way round also scores: "the angle at the circumference is half the angle at the centre".

Key idea

The diagram can look different. The lines may cross to make an arrowhead shape, or O may sit outside triangle ABC. The rule still works, as long as both angles stand on the same arc AB.

Double It, or Halve It

The 2-mark question: work out one angle, then give the reason. One mark for the angle, one for the reason.

Double it. A, B and C are on a circle, centre O. $∠ A C B = 3 1^{\circ}$ . Find angle AOB.

Worked example

1
$∠ A C B = 3 1^{\circ}$ — Given. C is on the circle, so this is the angle at the circumference. It stands on arc AB.
2
$∠ A O B = 2 \times 3 1^{\circ}$ — O is the centre, and its angle stands on the same arc AB. The angle at the centre is twice the angle at the circumference, so double it.
3
$∠ A O B = 6 2^{\circ}$ — 2 × 31 = 62. Write the reason next to it to earn the second mark.

Halve it. $∠ A O B = 14 0^{\circ}$ . Find angle ACB.

Worked example

1
$∠ A O B = 14 0^{\circ}$ — Given. This is the angle at the centre, standing on arc AB.
2
$∠ A C B = 14 0^{\circ} \div 2$ — C stands on the same arc AB. The angle at the circumference is half the angle at the centre, so halve it.
3
$∠ A C B = 7 0^{\circ}$ — 140 ÷ 2 = 70. Then write the reason.

Rule

Reason: angle at the centre is twice the angle at the circumference.

Common mistake

Halving when you should double. Start at the circumference: double. Start at the centre: halve.

The Reflex Angle at the Centre

The 2-mark question may ask for the reflex angle at O. A reflex angle is bigger than 180°: it is the outside of the angle.

Worked example

1
$∠ A C B = 4 8^{\circ}$ — Given. C is on the circle, so this is the angle at the circumference, standing on arc AB.
2
$∠ A O B = 2 \times 4 8^{\circ} = 9 6^{\circ}$ — Angle at the centre is twice the angle at the circumference. This is the small angle at O, under 180°.
3
$reflex ∠ A O B = 36 0^{\circ} - 9 6^{\circ}$ — The reflex angle is the rest of the full turn round O. Angles at a point add up to 360°.
4
$reflex ∠ A O B = 26 4^{\circ}$ — 360 − 96 = 264. It is more than 180°, so it is reflex, as the question asked.

Common mistake

Stopping at $9 6^{\circ}$ . The question asked for the reflex angle, so take it away from $36 0^{\circ}$ .

An Obtuse Angle on the Circle

If the angle on the circle is more than 90°, it stands on the big arc. Its angle at the centre is then the reflex angle at O.

P, Q and R are on a circle, centre O. $∠ P QR = 11 8^{\circ}$ . Find angle POR, the one less than 180°.

Worked example

1
$∠ P QR = 11 8^{\circ}$ — Given. It is more than 90°, so Q sits on the small arc and the angle stands on the big arc PR.
2
$reflex ∠ P O R = 2 \times 11 8^{\circ}$ — The angle at O on the same big arc is the reflex one. Angle at the centre is twice the angle at the circumference.
3
$reflex ∠ P O R = 23 6^{\circ}$ — 2 × 118 = 236. It is more than 180°, as a reflex angle must be.
4
$∠ P O R = 36 0^{\circ} - 23 6^{\circ}$ — The question wants the other angle at O. Angles at a point add up to 360°.
5
$∠ P O R = 12 4^{\circ}$ — 360 − 236 = 124. It is less than 180°, so it is the angle asked for.

Common mistake

Giving 236° as angle POR. That is the reflex angle. Take it away from 360° when the question wants the angle under 180°.

Key idea

Check another way: pick a point S on the big arc. Opposite angles of a cyclic quadrilateral add up to 180°, so $∠ P S R = 18 0^{\circ} - 11 8^{\circ} = 6 2^{\circ}$ , and $2 \times 6 2^{\circ} = 12 4^{\circ}$ . Same answer.

Radii First, Then Halve

Sometimes the angle you are given is where a radius meets a chord, not an angle at C. Two radii make an isosceles triangle, so use that first, then halve.

A, B and C are on a circle, centre O. C and O are on the same side of AB. $∠ O A B = 3 4^{\circ}$ . Find angle ACB.

Worked example

1
$∠ O A B = 3 4^{\circ}$ — Given. OA and OB are both radii, so triangle OAB is isosceles.
2
$∠ O B A = 3 4^{\circ}$ — Base angles of an isosceles triangle are equal. They sit opposite the equal sides OA and OB.
3
$∠ A O B = 18 0^{\circ} - 3 4^{\circ} - 3 4^{\circ}$ — Angles in a triangle add up to 180°. This finds the angle at the centre.
4
$∠ A O B = 11 2^{\circ}$ — 180 − 68 = 112.
5
$∠ A C B = 11 2^{\circ} \div 2$ — C stands on the same arc AB. Angle at the centre is twice the angle at the circumference, so halve it.
6
$∠ A C B = 5 6^{\circ}$ — 112 ÷ 2 = 56.

Key idea

Write $∠ A O B = 11 2^{\circ}$ on the diagram. In a 2-mark question that line earns a mark on its own, even if the last step slips.

Common mistake

Saying the base angles are equal "because it looks isosceles". Give the real reason: OA and OB are radii, so they are equal.

Which Method? Read the Question

Look at where the angle sits: at O, or on the circle.

You see	You do
Angle at O, find the edge angle	halve it
Edge angle, find the angle at O	double it
Reflex angle at O	360° take away, then halve or double
Edge angle over 90°	double it: that is the reflex angle at O
Angle where a radius meets a chord	isosceles first, then halve
O and two radii, find a base angle	Combining Circle Theorems
Two edge angles on the same arc	Angles in the Same Segment
One side is a diameter	Angle in a Semicircle

Key idea

Check both angles start from the same two points A and B. If not, the doubling rule does not apply.

Practise angle at the centre now

Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.

Practise free on Lemmara

Real past-paper questions on angle at the centre

Each with a full worked solution.

All 10 questions on this topic →

Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.