# 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:**

We use the concept related to the properties of circles to solve the problem.

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

(i) Area of the sector = θ/360 × πr^{2}

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

Draw a figure to visualize the area to be found out.

Here, radius, r = 15 cm, θ = 60°

Visually it’s clear from the figure that;

AB is the chord 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 = πr^{2} - Area of minor segment APB

Here, Rradius, r = 15 cm, θ = 60°

Area of the sector OAPB = θ/360° × πr^{2}

= 60°/360° × 3.14 × 15 × 15 cm^{2}

= 117.75 cm^{2}

In ΔAOB,

OA = OB = r (radii of the circle)

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

∠AOB + ∠OBA + ∠OAB = 180° (Angle sum 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 r^{2}

= √3/4 × (15 cm)^{2}

= 1.73/4 × 225 cm^{2}

= 97.3125 cm^{2}

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

= 117.75 cm^{2} - 97.3125 cm^{2}

= 20.4375 cm^{2}

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

= π × (15 cm)^{2} - 20.4375 cm^{2}

= 3.14 × 225 cm^{2} - 20.4375 cm^{2}

= 706.5 cm^{2} - 20.4375 cm^{2}

= 686.0625 cm^{2}

**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:

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

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 cm^{2} and 686.0625 cm^{2} respectively