Let’s draw the diagram.
Let the side of the square be $x.$ And
DIAGRAM
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 $= \frac{\pi \; x^{2}}{4} : \frac{\pi \; x^{2}}{2} = \frac{1}{4} : \frac{1}{2} = 1:2$
Correct Answer $: \text{D}$