Find the area of the segment of a circle of radius 12 cm whose corresponding sector has a central angle of 60° (Use π = 3.14)

Solution:

Given, radius of circle, r = 12 cm

Central angle, θ = 60° 

We have to find the area of the segment of a circle.

Find the area of the segment of a circle of radius 12 cm whose corresponding sector has a central angle of 60° (Use π = 3.14)

From the figure,

OA = OB = 12 cm

∠AOB = 60° 

So, AOB is an isosceles triangle.

Two sides OA and OB are equal.

We know that the angles opposite to equal sides are equal.

So, ∠OAB = ∠OBA 

Let ∠OAB = ∠OBA = θ

We know that the sum of all three interior angles of a triangle is always equal t 180°

∠AOB + ∠OAB + ∠OBA = 180°

60° + θ + θ = 180°

60° + 2θ = 180°

2θ = 180° - 60° 

2θ = 120°

θ = 120°/2

θ = 60°

So, ∠OAB = ∠OBA = 60°

All the three interior angles are equal to 60°

The sides OA = OB = AB = 12 cm

Thus, AOB is an equilateral triangle.

Area of the segment = area of the sector - area of triangle

Area of sector = πr²θ/360°

= (3.14)(12)²(60°/360°)

= (3.14)(12)²(1/6)

= (3.14)(2)(12)

= 24(3.14)

= 75.36 cm²

Area of equilateral triangle AOB = (√3/4)a²

= (√3/4)(12)²

= (√3)(3)(12)

= 36√3

= 62.354 cm²

Area of segment = 75.36 - 62.354

= 13.006 cm²

Therefore, the area of the segment is 13.006 cm²

✦ Try This: Find the area of the segment of a circle of radius 16 cm whose corresponding sector has a central angle of 120° (Use π = 3.14).

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 12

NCERT Exemplar Class 10 Maths Exercise 11.4 Problem 4

Find the area of the segment of a circle of radius 12 cm whose corresponding sector has a central angle of 60° (Use π = 3.14)

Summary:

The area of the segment of a circle of radius 12 cm whose corresponding sector has a central angle of 60° is 13.006 cm²

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