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

Let the side of the square be x. Then, its diagonal =  $2x2=2x$ 

Radius of incircle =  $\frac{x}{2}$ 

Radius of circum circle=  $2\times \frac{x}{2}=\frac{x}{2}$ 

Required ratio =  $\frac{\mathrm{\pi x}}{2}:\frac{\mathrm{\pi x}}{2}=\frac{1}{4}:\frac{1}{2}=1:2$