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

Areas Related to Circles | Exercise 11.1

Question 3

The length of the minute hand of a clock is $14\text{ cm}$ . Find the area swept by the minute hand in $5$ minutes.

Solution

Understand the Question

The minute hand of a clock acts as the radius of a circular region, so $r = 14\text{ cm}$ .
A complete rotation of the minute hand covers $360^\circ$ in $60\text{ minutes}$ .
The region swept by the minute hand in $5\text{ minutes}$ forms a circular sector.
The area of a sector with central angle $\theta$ and radius $r$ is given by: $\text{Area} = \dfrac{\theta}{360^\circ} \times \pi r^2$

Step 1 · Find the Angle Swept in 5 Minutes

A minute hand completes $360^\circ$ in $60\text{ minutes}$ . Diagram 1

Angle swept in $1\text{ minute}$ : $\dfrac{360^\circ}{60} = 6^\circ$

Angle swept in $5\text{ minutes}$ ( $\theta$ ): $\theta = 5 \times 6^\circ = 30^\circ$

Step 2 · Calculate the Area of the Sector

Radius $r = 14\text{ cm}$ and $\theta = 30^\circ$ .

\begin{aligned} \text{Area} &= \dfrac{\theta}{360^\circ} \times \pi r^2 \\[0.6em] &= \dfrac{30^\circ}{360^\circ} \times \pi (14)^2 \\[0.6em] &= \dfrac{1}{12} \times \dfrac{22}{7} \times 14 \times 14 \\[0.6em] &= \dfrac{1}{12} \times 22 \times 2 \times 14 \\[0.6em] &= \dfrac{1}{12} \times 616 \\[0.6em] &= \dfrac{154}{3} \end{aligned}

Answer

$\dfrac{154}{3}\text{ cm}^2$

Common Mistakes

Minute Hand vs Hour Hand: The minute hand rotates $360^\circ$ in $60\text{ minutes}$ ( $6^\circ\text{/min}$ ), whereas the hour hand rotates $360^\circ$ in $12\text{ hours}$ ( $0.5^\circ\text{/min}$ ). Do not mix up their rotation speeds.
Missing Units: Forgetting to append $\text{cm}^2$ to the final area.

More questions in Exercise 11.1

Unless stated otherwise, use $\pi = \dfrac{22}{7}$ .

Find the area of a sector of a circle with radius $6 \text{ cm}$ if angle of the sector is $60^\circ$ .

Find the area of a quadrant of a circle whose circumference is $22\text{ cm}$ .

The length of the minute hand of a clock is $14\text{ cm}$ . Find the area swept by the minute hand in $5$ minutes.

A chord of a circle of radius $10\text{ cm}$ subtends a right angle at the centre. Find the area of the corresponding :

(i) minor segment

(ii) major sector. (Use $\pi = 3.14$ )

In a circle of radius $21 \text{ cm}$ , an arc subtends an angle of $60^\circ$ at the centre. Find:

(i) the length of the arc

(ii) area of the sector formed by the arc

(iii) area of the segment formed by the corresponding chord

A chord of a circle of radius $15\text{ cm}$ subtends an angle of $60^\circ$ at the centre. Find the areas of the corresponding minor and major segments of the circle.

(Use $\pi = 3.14$ and $\sqrt{3} = 1.73$ )

A chord of a circle of radius $12\text{ cm}$ subtends an angle of $120^\circ$ at the centre. Find the area of the corresponding segment of the circle.

(Use $\pi = 3.14$ and $\sqrt{3} = 1.73$ )

A horse is tied to a peg at one corner of a square shaped grass field of side $15\text{ m}$ by means of a $5\text{ m}$ long rope (see Fig. 11.8). Find

(i) the area of that part of the field in which the horse can graze.

(ii) the increase in the grazing area if the rope were $10\text{ m}$ long instead of $5\text{ m}$ . (Use $\pi = 3.14$ )

A brooch is made with silver wire in the form of a circle with diameter $35\text{ mm}$ . The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 11.9. Find :

(i) the total length of the silver wire required.

(ii) the area of each sector of the brooch.

Q10

An umbrella has 8 ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius $45\text{ cm}$ , find the area between the two consecutive ribs of the umbrella.

Q11

A car has two wipers which do not overlap. Each wiper has a blade of length $25\text{ cm}$ sweeping through an angle of $115^\circ$ . Find the total area cleaned at each sweep of the blades.

Q12

To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle $80^\circ$ to a distance of $16.5\text{ km}$ . Find the area of the sea over which the ships are warned. (Use $\pi = 3.14$ )

Q13

A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is $28\text{ cm}$ , find the cost of making the designs at the rate of ₹ $0.35$ per $\text{cm}^2$ . (Use $\sqrt{3} = 1.7$ )

Q14

Tick the correct answer in the following :

Area of a sector of angle $p$ (in degrees) of a circle with radius $R$ is

(A) $\dfrac{p}{180} \times 2\pi R$ (B) $\dfrac{p}{180} \times \pi R^2$ (C) $\dfrac{p}{360} \times 2\pi R$ (D) $\dfrac{p}{720} \times 2\pi R^2$