Question

Find the ratio of the area of the circle circumscribing a square to the area of the circle inscribed in the square .

Answer 1

Let the side of the square inscribed in a square be a units.
Diameter of the circle outside the square = Diagonal of the square = $\sqrt{2}a$
Radius = $\frac{\sqrt{2}a}{2}=\frac{a}{\sqrt{2}}$
So, the area of the circle circumscribing the square = $\mathrm{\pi }{\left(\frac{\mathrm{a}}{\sqrt{2}}\right)}^2$ .....(i)
Now, the radius of the circle inscribed in a square = $\frac{a}{2}$
Hence, area of the circle inscribed in a square = $\mathrm{\pi }{\left(\frac{\mathrm{a}}{2}\right)}^2$ .....(ii)
From (i) and (ii)
$$\begin{gathered} \frac{\mathrm{Area}\mathrm{of}\mathrm{circle}\mathrm{circumscribing}\mathrm{a}\mathrm{square}}{\mathrm{Area}\mathrm{of}\mathrm{circle}\mathrm{inscribed}\mathrm{in}\mathrm{a}\mathrm{square}}=\frac{\mathrm{\pi }{\left({\displaystyle \frac{\mathrm{a}}{\sqrt{2}}}\right)}^2}{\mathrm{\pi }{\left({\displaystyle \frac{\mathrm{a}}{2}}\right)}^2} \\ =\frac{{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{1}{4}}} \\ =\frac{2}{1} \end{gathered}$$
Hence, the required ratio is 2 : 1.