As we know that area of the circle is directly proportional to the square of its radius.
Hence $$frac{A_{ic}}{A_{cc}} = frac{frac{x^2}{4}}{frac{x^2}{2}}$$
Where $$x$$ is side of square (say), ic is inscribed circle with radius $$frac{x}{2}$$, cc is circumscribed circle with radius $$frac{x}{sqrt{2}}$$
So ratio will be 1:2