# A round table cover has six equal designs as shown in Fig. 12.14. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs. 0.35 per cm^{2}. (Use √3 = 1.7)

**Solution:**

We use the concept of areas of circles and the area of hexagons to solve the problem given.

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

Visually it is clear that the designs are segments of the circle

∴ Area of the design = Area of 6 segments of the circle.

Since the table cover has 6 equal design, therefore angle of each segment at the center = 360°/6 = 60°

Consider segment APB. Chord AB subtends an angle of 60° at the center.

∴ Area of segment APB = Area of sector AOPB - Area of ΔAOB

From the figure, we observe the designs are made in the segments of a circle.

Since the table cover has 6 equal designs

∴ the angle subtended by each chord (bounding the segment) at the centre = 360°/6 = 60°

Consider ΔAOB

OB = OA

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

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

2 ∠OAB = 180° - 60°

∠OAB = 120°/2 = 60°

∴ ΔAOB is an equilateral triangle

Area of ΔAOB = √3/4 (side)^{2}

= √3/4 (28)^{2}

= √3 × 7 × 28

= 196√3

= 196 × 1.7

= 333.2 cm^{2}

Area of sector OAPB = 60°/360° × πr^{2}

= 1/6 × 22/7 × 28 × 28

= (11 × 4 × 28)/3

= 1232/3 cm^{2}

Area of segment APB = Area of sector OAPB - Area of ΔAOB

= (1232/3 - 333.2) cm^{2}

Area of designs = 6 Area of segment

= 6 × (1232/3 - 333.2) cm^{2}

= 2464 - 1999.2 cm^{2}

= 464.8 cm^{2}

Cost of making 1 cm^{2} of designs = ₹ 0.35

∴ Cost of making 464.8 cm^{2} of designs

= ₹ 0.35 × 464.8

= ₹ 162.68

**Video Solution:**

## A round table cover has six equal designs as shown in Fig. 12.14. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs. 0.35 per cm^{2}

### NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.2 Question 13 - Chapter 12 Exercise 12.2 Question 13

A round table cover has six equal designs as shown in Fig. 12.14. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs. 0.35 per cm^{2}

The cost of making 464.8 cm^{2} area of designs of a round table cover is ₹ 162.68