5.2 Radian measure, arc length and sector area — Practise

Work through these on your own. Check the unit of every angle before you use it. Leave answers in terms of $\pi$ when the question asks for an exact value, and give other answers to three significant figures.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Change $135^\circ$ into radians, in terms of $\pi$ .
Change $\dfrac{5\pi}{6}$ radians into degrees.
Change $2.4$ radians into degrees, to one decimal place.
A circle has radius $12$ cm. Find the length of the arc that subtends $0.6$ radians at the centre.
A sector has radius $9$ cm and angle $2$ radians. Find its area.
A sector has radius $10$ cm and angle $\dfrac{\pi}{4}$ radians. Find the arc length and the sector area, in terms of $\pi$ .
A circle has radius $5$ cm. Find the area of the segment cut off by an arc of angle $\dfrac{\pi}{6}$ radians. Give the exact value.
Change $210^\circ$ into radians, in terms of $\pi$ .
A sector has radius $8$ cm and arc length $20$ cm. Find the angle at the centre, in radians.
A sector has radius $6$ cm and area $30$ cm $^2$ . Find the angle at the centre, in radians.

Answers and concise solutions

1. The angle in radians

Answer: Multiply by $\dfrac{\pi}{180}$ : $135 \times \dfrac{\pi}{180} = \dfrac{135\pi}{180} = \dfrac{3\pi}{4}$ .

2. The angle in degrees

Answer: Multiply by $\dfrac{180}{\pi}$ : $\dfrac{5\pi}{6} \times \dfrac{180}{\pi} = \dfrac{5 \times 180}{6} = 150^\circ$ .

3. The angle in degrees

Answer: $2.4 \times \dfrac{180}{\pi} \approx 137.5^\circ$ .

4. The arc length

Answer: The angle is already in radians, so $l = r\theta = 12 \times 0.6 = 7.2$ cm.

5. The sector area

Answer: $A = \tfrac{1}{2}r^2\theta = \tfrac{1}{2} \times 81 \times 2 = 81$ cm $^2$ .

6. The arc length and the area

Answer: Arc: $l = 10 \times \dfrac{\pi}{4} = \dfrac{5\pi}{2}$ cm. Area: $A = \tfrac{1}{2} \times 100 \times \dfrac{\pi}{4} = \dfrac{25\pi}{2}$ cm $^2$ .

7. The segment area

Answer: Sector $= \tfrac{1}{2} \times 25 \times \dfrac{\pi}{6} = \dfrac{25\pi}{12}$ . Triangle $= \tfrac{1}{2} \times 25 \times \sin\dfrac{\pi}{6} = \dfrac{25}{4}$ . Segment $= \dfrac{25\pi}{12} - \dfrac{25}{4} \approx 0.295$ cm $^2$ .

8. The angle in radians

Answer: $210 \times \dfrac{\pi}{180} = \dfrac{210\pi}{180} = \dfrac{7\pi}{6}$ .

9. The angle in radians

Answer: Rearrange $l = r\theta$ : $\theta = \dfrac{l}{r} = \dfrac{20}{8} = 2.5$ radians.

10. The angle in radians

Answer: Rearrange $A = \tfrac{1}{2}r^2\theta$ : $\theta = \dfrac{2A}{r^2} = \dfrac{60}{36} = \dfrac{5}{3}$ radians.