Areas Related to Circles | Exercise 11.1

Question 5

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

Solution

Understand the Question

A circle has radius $r = 21\text{ cm}$ and an arc subtending a central angle $\theta = 60^\circ$ .
Arc Length: Given by $L = \dfrac{\theta}{360^\circ} \times 2\pi r$ .
Area of Sector: Given by $A_{\text{sector}} = \dfrac{\theta}{360^\circ} \times \pi r^2$ .
Area of Segment: The segment area is found by subtracting the area of $\Delta OAB$ from the sector area: $\text{Area of Segment} = \text{Area of Sector} - \text{Area of } \Delta OAB$ Since $OA = OB = 21\text{ cm}$ and $\angle AOB = 60^\circ$ , $\Delta OAB$ is an equilateral triangle with area $\dfrac{\sqrt{3}}{4} \times (\text{side})^2$ .

(i) Find the length of the arc

Step 1 · Calculate Arc Length

Given radius $r = 21\text{ cm}$ and angle $\theta = 60^\circ$ . Diagram 1

\begin{aligned} L &= \dfrac{\theta}{360^\circ} \times 2\pi r \\[0.6em] &= \dfrac{60^\circ}{360^\circ} \times 2 \times \dfrac{22}{7} \times 21 \\[0.6em] &= \dfrac{1}{6} \times 2 \times 22 \times 3 \\[0.6em] &= 22 \end{aligned}

Answer

(i) $22\text{ cm}$

(ii) Find the area of the sector formed by the arc

Step 1 · Calculate Sector Area

\begin{aligned} A_{\text{sector}} &= \dfrac{\theta}{360^\circ} \times \pi r^2 \\[0.6em] &= \dfrac{60^\circ}{360^\circ} \times \dfrac{22}{7} \times (21)^2 \\[0.6em] &= \dfrac{1}{6} \times \dfrac{22}{7} \times 21 \times 21 \\[0.6em] &= 231 \end{aligned}

Answer

(ii) $231\text{ cm}^2$

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

Step 1 · Calculate Segment Area

In $\Delta OAB$ , $OA = OB = 21\text{ cm}$ and $\angle AOB = 60^\circ$ , so $\Delta OAB$ is an equilateral triangle.

\begin{aligned} A_{\text{triangle}} &= \dfrac{\sqrt{3}}{4} \times (21)^2 \\[0.6em] &= \dfrac{441\sqrt{3}}{4} \end{aligned}

\begin{aligned} A_{\text{segment}} &= A_{\text{sector}} - A_{\text{triangle}} \\[0.6em] &= 231 - \dfrac{441\sqrt{3}}{4} \end{aligned}

Answer

(iii) $\left(231 - \dfrac{441\sqrt{3}}{4}\right) \text{ cm}^2$

Common Mistakes

Equilateral Triangle Identification: Forgetting that an isosceles triangle with a $60^\circ$ vertex angle is equilateral, with area given by $\dfrac{\sqrt{3}}{4}a^2$ .
Formula Confusion: Swapping the arc length formula (which uses $2\pi r$ ) with the sector area formula (which uses $\pi r^2$ ).
Segment vs. Sector: Subtracting the triangle area is necessary to find the segment area; it is not the same as the sector 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) $\dfrac{p}{720} \times 2\pi R^2$