# 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 × πr^{2}

(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 = 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 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 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 major 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}

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

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

**☛ Related Questions:**

- 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)
- A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig. 12.11). Find(i) the area of that part of the field in which the horse can graze.(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m.(Use π = 3.14)
- A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 12.12. Find :(i) the total length of the silver wire required.(ii) the area of each sector of the brooch.
- An umbrella has 8 ribs which are equally spaced (see Fig. 12.13). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

visual curriculum