NCERT Class X Chapter 11: Area Related To Circles Exercise 11.1 Question 14

NCERT Class X Chapter 12: Areas Related to Circles

Question:

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

Given:

Radius of the circle \( = R \)
Angle of the sector \( = p^\circ \)

To Find:

Area of the sector of the circle.

Formula:

Area of a sector of angle \( \theta^\circ \) and radius \( r \) is:

$$ \text{Area} = \dfrac{\theta}{360} \times \pi r^2 $$

Solution:

Step 1: Area of a complete circle of radius \( R \) is:

$$ \pi R^2 $$

Step 2: A complete circle corresponds to an angle of:

$$ 360^\circ $$

Step 3: Area of a sector of angle \( p^\circ \) is:

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

Step 4: Rewriting the expression:

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

Result:

The area of the sector is:

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

Hence, the correct option is (D).

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