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 Class 9

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 Class 9

Question 1.

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 $\frac{2}{\pi} \approx 0.637$. Class 9

Solution:

A square is inscribed in a circle of radius r ... [Given]

∴ Diagonal of square = Diameter of circle = 2r

∵ In a square, diagonal = side × $\sqrt{2}$

$\begin{array}{ll}\therefore \quad & \mathrm{s} \sqrt{2}=2 r \\ & \mathrm{~s}=\frac{2 r}{\sqrt{2}}=\sqrt{2} r\end{array}$

∵ Area of square = $s^2=(\sqrt{2} r)^2=2 r^2$

∵ Area of circle = πr²

∴ Required ratio

$\begin{aligned} & =\frac{\text { Area of square }}{\text { Area of circle }} \\ & =\frac{2 r^2}{\pi r^2}=\frac{2}{\pi} \approx 0.637\end{aligned}$

∴ Ratio of the area of the square to area of circle = $\frac{2}{\pi} \approx 0.637$