What is the ratio between area of circles circumscribing a square and that of the one inscribed in the square?

Let the side length of the square be $a$.
1. Inscribed Circle (Inner Circle):
The diameter of the inscribed circle is equal to the side length of the square.

$$\text{Diameter}=a⟹\text{Radius }(r)=\frac{a}{2}$$

$$\text{Area of inscribed circle }(A_{\text{in}})=\pi r^2=\pi {\left(\frac{a}{2}\right)}^2=\frac{\pi a^2}{4}$$

2. Circumscribing Circle (Outer Circle):
The diameter of the circumscribing circle is equal to the diagonal of the square.

$$\text{Diagonal}=a\sqrt{2}$$

$$\text{Diameter}=a\sqrt{2}⟹\text{Radius }(R)=\frac{a\sqrt{2}}{2}=\frac{a}{\sqrt{2}}$$

$$\text{Area of circumscribing circle }(A_{\text{out}})=\pi R^2=\pi {\left(\frac{a}{\sqrt{2}}\right)}^2=\frac{\pi a^2}{2}$$

Now, finding the ratio of the area of the circumscribing circle to the inscribed circle:

$$\text{Ratio}=\frac{A_{\text{out}}}{A_{\text{in}}}=\frac{\frac{\pi a^2}{2}}{\frac{\pi a^2}{4}}=\frac{1/2}{1/4}=2:1$$