NCERT Class X Chapter 11: Area Related To Circles Exercise 11.1 Question 7

NCERT Class X Chapter 12: Areas Related to Circles

Question:

A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle.
(Use π = 3.14 and √3 = 1.73)

Given:

Radius of the circle, \( r = 12 \) cm
Angle subtended at the centre, \( \theta = 120^\circ \)

To Find:

Area of the corresponding segment of the circle

Formula:

Area of segment = Area of sector − Area of triangle

$$ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 $$ $$ \text{Area of triangle} = \frac{1}{2} r^2 \sin \theta $$

Solution:

Step 1: Find the area of the sector of the circle.

$$ \text{Area of sector} = \frac{120}{360} \times 3.14 \times 12^2 = \frac{1}{3} \times 3.14 \times 144 = 150.72 \text{ cm}^2 $$

Step 2: Find the area of the triangle formed by the radii and the chord.

$$ \sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} = \frac{1.73}{2} $$ $$ \text{Area of triangle} = \frac{1}{2} \times 12^2 \times \frac{1.73}{2} = 36 \times 1.73 = 62.28 \text{ cm}^2 $$

Step 3: Find the area of the corresponding segment.

$$ \text{Area of segment} = 150.72 - 62.28 = 88.44 \text{ cm}^2 $$

Result:

The area of the corresponding segment of the circle is 88.44 cm2.

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