A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

the radius of inscribed circle be $r$ and the radius of circumcircle be $\text{R}.$

Now, $r=\frac{x}{2}$

The Area of inscribed circle $= \pi \; r^{2} = \pi \; \left(\frac{x}{2}\right)^{2} = \frac{\pi \; x^{2}}{4} $

The diagonal of the square will be the diameter of the circumscribed circle.

So, $\text{R} = \frac{\sqrt{2} \; x}{2}$

The area of the circumscribed circle $= \pi \; \text{R}^{2} = \pi \; (\frac{\sqrt{2} \; x}{2})^{2} = \frac{\pi \; 2 x^{2}}{4} = \frac{\pi \; x^{2}}{2}$

The required ratio $= \frac{\pi \; x^{2}}{4} : \frac{\pi \; x^{2}}{2} = \frac{1}{4} : \frac{1}{2} = 1:2$