In Fig., 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.

Let us given that,

ABCD is a square of side 14 cm and here 4 circles are formed.

Since the circles are touching each other extremely.

Consider, the radius of each circle is r

and ABCD is a square so each angle is 90⁰.

So, the angles and radii for all sectors are equal.

We know that, the formula for the area of the sector of a circle.

Area of the sector = $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

= $\frac{{90}^0}{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

= ${\mathrm{\pi r}}^2$/4 cm²

Radius = 14/2 = 7 cm

Area of each sector = 1/4 $\times$ 22/7 $\times$ 7 $\times$7 

= 77/2 cm²

Area of shaded region = Area of square $-$ 4 $\times$ Area of each sector

= (14)² $-$ 4 $\times$77/2 

= 196 $-$ 154 

= 42 cm²