A-level · Trigonometry

Arc length and sector areaExam-style questions

The radian formulas for arc length, sector area and segment area, where they come from, and using them in reverse — finding a radius or angle from a given perimeter or area, and the perimeters of sectors and segments.

Start practising Read the lesson

33 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The diagram shows a sector $OAB$ of a circle of radius $8\text{ cm}$ , where the angle $AOB = 0.8$ radians. Find
(a) the length of the arc $AB$ ,
(b) the area of the sector.
Level 2 · 4 marksTry it
2
The diagram shows a sector $OAB$ of a circle of radius $6\text{ cm}$ , in which the arc $AB$ has length $15\text{ cm}$ . Find
(a) the angle $AOB$ in radians,
(b) the area of the sector.
Level 2 · 4 marksTry it
3
The diagram shows a sector $OAB$ of a circle of radius $12\text{ cm}$ with angle $AOB = 75^\circ$ . Find the exact length of the arc $AB$ .
Level 2 · 2 marksTry it
4
A sector $OAB$ of a circle with centre $O$ has radius $r$ cm and angle $\theta$ radians. The arc $AB$ has length $12$ cm and the sector has area $36$ cm $^2$ .
(a) Find $r$ and $\theta$ . (4 marks)
(b) Find the area of the segment bounded by the chord $AB$ and the arc $AB$ , giving your answer to 3 significant figures. (2 marks)
Level 2 · 6 marksTry it
5
A sector $OPQ$ of a circle has centre $O$ and radius $r$ cm, where angle $POQ = 0.9$ radians.

The area of the sector $OPQ$ is $115.2$ cm $^2$ .
(a) Show that $r = 16$ .

(2 marks)
(b) Find the perimeter of the sector $OPQ$ .

(2 marks)
Level 2 · 4 marksTry it