5.2 Radian measure, arc length and sector area — Practise with help

Before each question, write down whether the angle is in degrees or radians. The arc and sector formulas need radians, so convert first when you need to.

Warm-up: remember

Work out the circumference of a circle of radius $7$ cm, in terms of $\pi$ .
Work out the area of the same circle, in terms of $\pi$ .
Simplify $\dfrac{120}{180}$ .

Answers

$C = 2\pi r = 14\pi$ cm.
$A = \pi r^2 = 49\pi$ cm $^2$ .
$\dfrac{120}{180} = \dfrac{2}{3}$ .

If question 3 was hard, revise simplifying fractions before you continue.

Practise with help

1. Change $135^\circ$ into radians, in terms of $\pi$ .

Hint

Multiply the number of degrees by $\dfrac{\pi}{180}$ .

Next step

Write $135 \times \dfrac{\pi}{180}$ and simplify the fraction $\dfrac{135}{180}$ .

Full working

135 \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}.

2. Change $\dfrac{5\pi}{6}$ radians into degrees.

Hint

Multiply by $\dfrac{180}{\pi}$ . The $\pi$ cancels.

Next step

Work out $\dfrac{5}{6} \times 180$ .

Full working

\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150^\circ.

3. A circle has radius $12$ cm. Find the length of the arc that subtends $0.6$ radians at the centre.

Hint

The angle is already in radians, so use $l = r\theta$ directly.

Next step

Substitute $r = 12$ and $\theta = 0.6$ .

Full working

l = r\theta = 12 \times 0.6 = 7.2 \text{ cm}.

4. A sector has radius $9$ cm and angle $2$ radians. Find its area.

Hint

Use $A = \tfrac{1}{2}r^2\theta$ with the angle in radians.

Next step

The radius squared is $81$ . Work out $\tfrac{1}{2} \times 81 \times 2$ .

Full working

A = \tfrac{1}{2} \times 9^2 \times 2 = \tfrac{1}{2} \times 81 \times 2 = 81 \text{ cm}^2.

5. A sector has radius $10$ cm and angle $\dfrac{\pi}{4}$ radians. Find the length of the arc and the area of the sector.

Hint

This is two jobs. Use $l = r\theta$ first, then $A = \tfrac{1}{2}r^2\theta$ .

Next step

For the arc, work out $10 \times \dfrac{\pi}{4}$ . For the area, work out $\tfrac{1}{2} \times 100 \times \dfrac{\pi}{4}$ .

Full working

l = 10 \times \frac{\pi}{4} = \frac{5\pi}{2} \approx 7.85 \text{ cm}.

A = \tfrac{1}{2} \times 100 \times \frac{\pi}{4} = \frac{25\pi}{2} \approx 39.3 \text{ cm}^2.

6. A circle has radius $5$ cm. Find the area of the segment cut off by an arc of angle $\dfrac{\pi}{6}$ radians.

Hint

Find the sector, then subtract the triangle formed by the two radii. The triangle uses $\tfrac{1}{2}r^2\sin\theta$ .

Next step

Sector $= \tfrac{1}{2} \times 25 \times \dfrac{\pi}{6}$ . Triangle $= \tfrac{1}{2} \times 25 \times \sin\dfrac{\pi}{6}$ . Subtract.

Full working

\text{sector} = \frac{25\pi}{12} \approx 6.545.

\text{triangle} = \tfrac{1}{2} \times 25 \times \tfrac{1}{2} = 6.25.

\text{segment} = \frac{25\pi}{12} - 6.25 \approx 0.295 \text{ cm}^2.