A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.73)

Solution:

In a circle with radius r and the angle at the centre of degree measure θ,

(i) Area of the sector = θ/360 × πr²

(ii) Area of the segment = Area of the sector - Area of the corresponding triangle

Let's draw a figure to visualize the area to be calculated.

Here, radius, r = 15 cm, θ = 60°

Visually it’s clear from the figure that,

AB is the chord that subtends 60° angle at the centre.

(i) Area of minor segment APB = Area of sector OAPB - Area of ΔAOB

(ii) Area of major segment AQB = πr² - Area of minor segment APB

Here, Rradius, r = 15 cm, θ = 60°

Area of the sector OAPB = θ/360° × πr²

= 60°/360° × 3.14 × 15 × 15 cm²

= 117.75 cm²

In ΔAOB,

OA = OB = r (radii of the circle)

∠OBA = ∠OAB (Angles opposite to the equal sides in a triangle are equal)

∠AOB + ∠OBA + ∠OAB = 180° (Angle sum property of a triangle)

60° + ∠OAB + ∠OAB = 180° 

2 ∠OAB = 120°

∠OAB = 60°

∴ ΔAOB is an equilateral triangle because all its angles are equal.

⇒ AB = OA = OB = r

Area of ΔAOB = √3/4 × (side)²

= √3/4 r²

= √3/4 × (15 cm)²

= 1.73/4 × 225 cm²

= 97.3125 cm²

(i) Area of minor segment APB = Area of sector OAPB - Area of ΔAOB

= 117.75 cm² - 97.3125 cm²

= 20.4375 cm²

(ii) Area of the major segment AQB = Area of the circle - Area of minor segment APB

= π × (15 cm)² - 20.4375 cm²

= 3.14 × 225 cm² - 20.4375 cm²

= 706.5 cm² - 20.4375 cm²

= 686.0625 cm²

☛ Check: NCERT Solutions for Class 10 Maths Chapter 12

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Video Solution:

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle.

NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.2 Question 6

Summary:

The area of the minor segment APB and the area of the major segment AQB if a chord of a circle of radius 15 cm subtends an angle of 60° at the centre are 20.4375 cm² and 686.0625 cm² respectively.

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☛ Related Questions:

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