The ratio of the area of a circle and the maximum square inside it will:

Let the radius of the circle be $r$. Then the area of the circle is $\pi r^2$.The maximum square inscribed inside a circle has a diagonal length equal to the diameter of the circle ($d=2r$).Area of this square:
$${\text{Area}}_{\text{square}}=\frac{d^2}{2}=\frac{(2r{)}^2}{2}=\frac{4r^2}{2}=2r^2$$
Now find the ratio of their areas:
$$\text{Ratio}=\frac{{\text{Area}}_{\text{circle}}}{{\text{Area}}_{\text{square}}}=\frac{\pi r^2}{2r^2}=\frac{\pi }{2}$$
Since $\frac{\pi }{2}$ is a constant value, it does not depend upon the dimensions of the circle or square.