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 centers 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° = π/2 radians = θ (say)

Given that,

Radius of each sector = 7 cm

Area of the sector with central angle A = (½)r²θ

= ½ r² θ

= ½ × 49 × π/2

= ½ × 49 × (22/(2×7))

= (77/2) cm²

Since the central angles and the radius of each sector are same, area of each sector is 77/2 cm²

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

= 196 – (4 × (77/2))

= 196 – 154

= 42 cm²

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