Measuring Space: Perimeter and Area | Exercise 6.3

Question 3

The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes.

Solution

We need to find the area of the sector formed by the minute hand.

Step 1 — Find the angle swept

The minute hand completes a full circle in 60 minutes. A full circle is 360 degrees. Let's find the angle it sweeps in 1 minute.

$\text{Angle in 1 minute} = \frac{360^\circ}{60}$

$= 6^\circ$

Now, we find the angle swept in 10 minutes.

$\text{Angle in 10 minutes} = 10 \times 6^\circ$

$\boxed{60^\circ}$

Diagram 1

Step 2 — Calculate the area swept

The length of the minute hand is the radius of the sector. The radius $r$ is 7 cm. The angle $\theta$ swept is 60 degrees. The formula for the area of a sector is $(\theta/360^\circ) \times \pi r^2$ .

$\text{Area swept} = \frac{60^\circ}{360^\circ} \times \pi \times (7 \text{ cm})^2$

$= \frac{1}{6} \times \frac{22}{7} \times 49 \text{ cm}^2$

$= \frac{1}{6} \times 22 \times 7 \text{ cm}^2$

$= \frac{154}{6} \text{ cm}^2$

$= \frac{77}{3} \text{ cm}^2$

Answer

The area swept by the minute hand in 10 minutes is $\boxed{\frac{77}{3} \text{ cm}^2}$ .

More questions in Exercise 6.3

Unless stated otherwise, use the approximation $\frac{22}{7}$ for $\pi$ .

Find the area of a sector of a circle with radius 7 cm if the angle of the sector is 60°.

Find the area of a quadrant of a circle whose circumference is 44 cm.

The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes.

A chord of a circle of radius 10 cm subtends 90° at the centre. Find the area of the corresponding:

(i) minor sector (that subtends 90° at the centre), and (ii) major sector (that subtends 270° at the centre). (Use $\pi \approx 3.14$ .)

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use $\pi \approx 3.14$ and $\sqrt{3} \approx 1.73$ .)

A car has two wipers which do not overlap. Each wiper has a blade of length 28 cm and sweeps through an angle of 120°. Find the total area cleaned at each sweep of the blades.

A chord of a circle of radius $r$ subtends an angle of 60° at the centre of the circle. Show that the area of the corresponding minor segment of the circle is equal to $\pi r^2 \left( \frac{1}{6} - \frac{\sqrt{3}}{4} \right)$ .

An equilateral triangle is inscribed in a circle of radius $r$ . Show that the ratio of the area of the triangle to the area of the circle is equal to $\frac{3\sqrt{3}}{4\pi} \approx 0.413$ .

A square is inscribed in a circle of radius $r$ . Show that the ratio of the area of the square to the area of the circle is equal to $\frac{2}{\pi} \approx 0.637$ .

Q10

A hexagon is inscribed in a circle of radius $r$ . Show that the ratio of the area of the hexagon to the area of the circle is equal to $\frac{3\sqrt{3}}{2\pi} \approx 0.827$ . Can you see why the answer is exactly twice the answer to Question 8?