Find the ratio of the areas of the incircle and circumcircle of a square. · ABC26GN4290

Let the side of the square be $x$. Then, its diagonal $=\sqrt{2} x$. $$\begin{aligned} & \text { Radius of incircle }=\frac{x}{2} \text { and radius of circumcircle }=\frac{\sqrt{2} x}{2}=\frac{x}{\sqrt{2}} . \\ & \therefore \text { Required ratio }=\left(\frac{\pi x^{2}}{4}: \frac{\pi x^{2}}{2}\right)=\frac{1}{4}: \frac{1}{2}=1: 2 . \end{aligned}$$