In the figure,a square of diagonal $8\, cm$ is inscribed in a circle. Find the area of the shaded region.

(N/A) Let the side of the square be $a$ and the radius of the circle be $r$.
Given that,the length of the diagonal of the square $= 8\, cm$.
Since the diagonal of a square $= a\sqrt{2}$,we have:
$a\sqrt{2} = 8$
$a = \frac{8}{\sqrt{2}} = 4\sqrt{2}\, cm$.
When a square is inscribed in a circle,the diagonal of the square is equal to the diameter of the circle.
Therefore,the diameter of the circle $= 8\, cm$.
Radius of the circle $r = \frac{\text{Diameter}}{2} = \frac{8}{2} = 4\, cm$.
Area of the circle $= \pi r^2 = \pi(4)^2 = 16\pi\, cm^2$.
Area of the square $= a^2 = (4\sqrt{2})^2 = 16 \times 2 = 32\, cm^2$.
The area of the shaded region is the difference between the area of the circle and the area of the square:
Area of shaded region $= \text{Area of circle} - \text{Area of square}$
$= (16\pi - 32)\, cm^2$.
Hence,the required area of the shaded region is $(16\pi - 32)\, cm^2$.