In figure, ABCD is a square of side $14$ cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region.

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Solution

Area of shaded region
Given: Side of square ABCD = 14 cm
Radius of circles with centers A, B, C and D = 14/2 = 7 cm
Area of shaded region = Area of square - Area of four sectors subtending right angle
Area of each of the 4 sectors is equal to each other and is a sector of 90° in a circle of 7 cm radius. So, Area of four sectors will be equal to Area of one complete circle

So

Area of 4 sectors = Πr²

Area of square ABCD = (Side)²

Area of square ABCD = (14)²

Area of square ABCD = 196 cm²

Area of shaded portion = Area of square ABCD - 4 × Area of each sector
= 196 – 154
= 42 cm²

Therefore, the area of shaded portion is 42 cm²

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