A square is inscribed in a circle.

Let the side of the square be a and radius of the circle be r
We know that if a circle circumscribes a square, then the diameter of the circle is equal to the diagonal of the square.

$\therefore \sqrt{2} a=2 r$

$\Rightarrow a=\sqrt{2} r$

Now,

$\frac{\text { Area of circle }}{\text { Area of square }}=\frac{\pi r^{2}}{a^{2}}$

$=\frac{\pi r^{2}}{(\sqrt{2} r)^{2}}$

$=\frac{\pi r^{2}}{2 r^{2}}$

$=\frac{\pi}{2}$

Hence, the ratio of the areas of the circle and the square is π : 2