Question

In Fig. 13.113, ABCD is a square with side $2\sqrt{2}$ cm and inscribed in a circle. Find the area of the shaded region. (Use π = 3.14)

Answer 1

ABCD is a square.

Side of the square = AB = $2\sqrt{2}\mathrm{cm}$

We know

Length of the diagonal of square = $\sqrt{2}$ × Side of the square

∴ BD = Diameter of the circle = Length of the diagonal of square = $\sqrt{2}\times 2\sqrt{2}$ = 4 cm

⇒ Radius of the circle = $\frac{\mathrm{BD}}{2}$ = 2 cm

Now,

Area of shaded region

= Area of the circle − Area of the square ABCD

$$\begin{gathered} =\mathrm{\pi }{\left(\frac{\mathrm{BD}}{2}\right)}^2-{\left(\mathrm{AB}\right)}^2 \\ =3.14\times {\left(2\right)}^2-{\left(2\sqrt{2}\right)}^2 \\ =3.14\times 4-8 \end{gathered}$$
$$\begin{gathered} =12.56-8 \\ =4.56{\mathrm{cm}}^2 \end{gathered}$$

Thus, the area of the shaded region is 4.56 cm².