14. Tick the correct answer in the following : Area of a sector of angle (in degrees) of a circle with radius is (A) (B) (C) (D)

Understand the Question The area of a sector of angle (in degrees) in a circle of radius is given by the formula: Substituting , the required area is . We compare this formula with each option to find the matching or equivalent expression. Step 1 · Recall the Area of a Sector Formula The area of a sector of angle and radius is Step 2 · Check the Given Options (A) — incorrect (does not represent area). (B) — incorrect denominator. (C) — represents length of the arc, not area. (D) Simplifying Option (D) Option (D) matches the sector area formula. Answer > (D) Common Mistakes Confusing Area with Arc Length: Choosing option (C) , which is the formula for the length of the arc of the sector, not its area. Missing Equivalent Fractions: Rejecting option (D) at first glance because the denominator is rather than , without simplifying .

Areas Related to Circles | Exercise 11.1

Question 14

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) $\dfrac{p}{720} \times 2\pi R^2$

Solution

Understand the Question

The area of a sector of angle $\theta$ (in degrees) in a circle of radius $R$ is given by the formula: $\text{Area of sector} = \dfrac{\theta}{360} \times \pi R^2$
Substituting $\theta = p$ , the required area is $\dfrac{p}{360} \times \pi R^2$ .
We compare this formula with each option to find the matching or equivalent expression.

Step 1 · Recall the Area of a Sector Formula

The area of a sector of angle $p$ and radius $R$ is Diagram 1

$\text{Area} = \dfrac{p}{360} \times \pi R^2$

Step 2 · Check the Given Options

(A) $\dfrac{p}{180} \times 2\pi R$ — incorrect (does not represent area).
(B) $\dfrac{p}{180} \times \pi R^2$ — incorrect denominator.
(C) $\dfrac{p}{360} \times 2\pi R$ — represents length of the arc, not area.
(D) $\dfrac{p}{720} \times 2\pi R^2$

Simplifying Option (D)

\begin{aligned} \dfrac{p}{720} \times 2\pi R^2 &= \dfrac{p \times 2}{720} \times \pi R^2 \\[0.6em] &= \dfrac{p}{360} \times \pi R^2 \end{aligned}

Option (D) matches the sector area formula.

Answer

(D) $\dfrac{p}{720} \times 2\pi R^2$

Common Mistakes

Confusing Area with Arc Length: Choosing option (C) $\dfrac{p}{360} \times 2\pi R$ , which is the formula for the length of the arc of the sector, not its area.
Missing Equivalent Fractions: Rejecting option (D) at first glance because the denominator is $720$ rather than $360$ , without simplifying $\dfrac{2}{720} = \dfrac{1}{360}$ .

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) $\dfrac{p}{720} \times 2\pi R^2$