In the adjoining figure ABCD is a square. Four equal semicircles are drawn in such a way that they meet each other at O. AB, BC, CD and AD are diameters each equal to 8 cm. Find the area of the shaded region.

$32(\pi-2)$ sq. $\mathrm{cm}$ .

$16(\pi-2)$ sq. $\mathrm{cm}$ .

$(2 \pi-8)$ sq. cm.

$\left(\frac{3}{4} \pi-4\right)$ sq. $\mathrm{cm}$ .

Answer : Option A

Explanation :

Area of square = 64 sq. cm.

∴ 4(x + y) = 64

⇒ x + y = 16 .... (i)

In a semi-circle,

$\mathrm{AOB}=x+y+x=\frac{1}{2} \pi \times(4)^{2}$

⇒ 2x + y = 8π ....(ii)

From equations (i) and (ii)

x = 8S – 16

∴ Area of the shaded region

= 4 (8S – 16)

= 32 (S–2) sq. cm.