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.

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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
Figure 1

Diagram NOT accurately drawn

A paved seating area, $OPQR$ $O P QR$ , is shown in plan view in Figure 1.

The edge $PQ$ $P Q$ of the seating area is an arc of a circle with centre $O$ $O$ , and its other three edges are straight.

Given that
- $OP = OQ = 6$ m
- $OR = 7$ m and $QR = 4$ m
- angle $POR = \dfrac{\pi}{2}$ radians
(a) show that, to 3 decimal places, angle $QOR$ $QO R$ is $0.607$ $0.607$ radians.

(3 marks)
(b) Find, in m $^2$ $^{2}$ , the area of the seating area, giving your answer to 3 significant figures.

(3 marks)
(c) Find, in m, the perimeter of the seating area, giving your answer to 3 significant figures.

(3 marks)
Level 4 · 9 marksTry it
2
Figure 1

Diagram NOT accurately drawn

The plan view of a rose bed is shown in Figure 1.

The bed is made from two sectors with the same centre $O$ $O$ : the sector $OPQ$ $O P Q$ , and the sector $ORS$ $O R S$ of a larger circle. The point $Q$ $Q$ lies on $OR$ $O R$ .

Given that
- $P$ , $O$ and $S$ lie on a straight line
- angle $ROS = 0.9$ radians
- the arc $PQ$ has length $9.5$ m
- $QR = 2.5$ m
(a) show that $OR = 6.74$ $O R = 6.74$ m, correct to 3 significant figures.

(3 marks)
(b) Find, to 3 significant figures, the perimeter of the rose bed in metres.

(3 marks)
(c) Find the area of the rose bed, to the nearest square metre.

(3 marks)
Level 4 · 9 marksTry it
3
The diagram shows a sector $OAB$ of radius $r\text{ cm}$ and angle $\theta$ radians, with a fixed perimeter of $20\text{ cm}$ .
(a) Show that the area of the sector is $A = 10r - r^2$ .
(b) Hence find the value of $r$ that maximises the area, and the maximum area. (The diagram is not accurately drawn.)
Level 4 · 5 marksTry it
4
A sector of a circle has angle $1.2$ radians and perimeter $25.6$ cm.
(a) Show that the radius of the sector is $8$ cm.
(b) Find the area of the sector.
Level 4 · 5 marksTry it
5
A landscape designer is planning a flower bed in the shape of a sector of a circle with radius $r$ metres and angle $\theta$ radians. The perimeter of the bed must be exactly $40$ metres of edging.(a) Show that the area of the bed, $A$ m², satisfies

$A = 20r - r^2$ . (3 marks)
(b) Hence find the maximum possible area of the bed, justifying that your value is a maximum, and find the value of $\theta$ for which it occurs. (4 marks)
Level 4 · 7 marksTry it

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