Question

Four circular cardboard pieces of radii 7 cm are placed on a paper in such a way that each piece touches other two pieces. Find the area of the portion enclosed between these pieces.

Answer 1

According to the question,

The four circles are placed such that each piece touches the other two pieces.

By joining the centres of the circles by a line segment, we get a square ABDC with sides,

AB = BD = DC = CA = 2 ...(Radius)

= 2(7) cm

= 14 cm

Now, Area of the square = (Side)²

= (14)²

= 196 cm²

ABDC is a square,

Therefore, each angle has a measure of 90°.

i.e., ∠A = ∠B = ∠D = ∠C = 90° = `pi/2` radius = θ   ...(say)

Given that,

Radius of each sector = 7 cm

Area of the sector with central angle A = `(1/2)"r"^2θ`

= `1/2"r"^2θ`

= `1/2 xx 49 xx pi/2`

= `1/2 xx 49 xx 22/(2 xx 7)`

= `77/2 "cm"^2`

Since the central angles and the radius of each sector are same, area of each sector is `77/2 "cm"^2`

∴ Area of the shaded portion = Area of square – Area of the four sectors

= `196 - (4 xx 77/2)`

= 196 – 154

= 42 cm²

Therefore, the required area of the portion enclosed between these pieces is 42 cm².