A square is inscribed in a circle of radius $r$. Show that the ratio of the area of the square to the area of the circle is equal to $\frac{2}{\pi} \approx 0.637$.                                                                                                                                                     [Page no. 148]

For square inscribed in circle of radius $r$:

Diagonal of square = $2r$ (diameter of circle).

$$\text{If side of square} = a, \text{ then } a\sqrt{2} = 2r \Rightarrow a = r\sqrt{2}$$

$$\text{Area of square} = a^2 = 2r^2$$

$$\text{Area of circle} = \pi r^2$$

$\text{Ratio} = \frac{2r^2}{\pi r^2} = \frac{2}{\pi} \approx 0.637.$

Hence, proved.