Question

In Fig. 14.78, a circle is inscribed in a square ABCD and the square is circumscribed by a circle. If the radius of the smaller circle is r сm, then the area of the shaded region in cm² is _________.

Answer 1

Side of the square = Diameter of the smaller circle = 2r

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

∴ Radius of the bigger circle = $\frac{\mathrm{Length}\mathrm{of}\mathrm{diagonal}\mathrm{of}\mathrm{square}}{2}=\frac{2\sqrt{2}r}{2}=\sqrt{2}r$

Area of the shaded region

= $\frac{1}{4}$(Area of the bigger circle − Area of the square)

$$\begin{gathered} =\frac{1}{4}\left[\mathrm{\pi }{\left(\sqrt{2}r\right)}^2-{\left(2r\right)}^2\right] \\ =\frac{1}{4}\left(2\mathrm{\pi }{r}^2-4r^2\right) \\ =\left(\frac{\mathrm{\pi }-2}{2}\right)r^2{\mathrm{cm}}^2 \end{gathered}$$

In Fig. 14.78, a circle is inscribed in a square ABCD and the square is circumscribed by a circle. If the radius of the smaller circle is r сm, then the area of the shaded region in cm² is $\underline{\left(\frac{\mathrm{\pi }-2}{2}\right)r^2}$.