GCSE · Circles, arcs & sectors

Circle theoremsExam-style questions

The theorems about angles in a circle — at the centre, in a semicircle, in the same segment, and in a cyclic quadrilateral — why each is true, and the exact wording examiners require as a reason.

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16 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
$A$ , $B$ , $C$ and $D$ are points on a circle.

$AC$ is a diameter of the circle.

Angle $BAC = 27°$ .Diagram NOT accurately drawn

Work out the size of angle $ADB$ .

Give reasons for your answer. (4 marks)
Level 4 · 4 marksTry it
2
$A$ , $B$ , $C$ and $D$ are points on a circle, centre $O$ .Angle $AOC = 124°$

Work out the size of angle $ABC$ .

Write down any circle theorems you use.

(4 marks)
Level 4 · 4 marksTry it
3
The four vertices of rectangle $PQRS$ lie on a circle, centre $O$ . $PQ = 11$ cm $\qquad QR = 6.5$ cm

Work out the circumference of the circle.

Give your answer correct to 3 significant figures.

(4 marks)
Level 4 · 4 marksTry it
4
$A$ , $B$ , $C$ and $D$ are points on a circle with centre $O$ .

Angle $AOC = 130°$ , where $AOC$ is the angle at the centre standing on the arc $ABC$ .

$ABCD$ is a cyclic quadrilateral.

(a) Work out the size of angle $ADC$ . Give a reason for your answer.

(b) Work out the size of angle $ABC$ . Give a reason for your answer.
Level 4 · 6 marksTry it
5
$A$ , $B$ and $C$ are points on a circle with centre $O$ .

The reflex angle $AOB = 216°$ .Diagram NOT accurately drawn

Work out the size of the angle marked $x$ . Give a reason for your answer.
Level 4 · 3 marksTry it

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