jmc

Let $s_1$ be the side length of the square inscribed in the semicircle of radius $r$. Applying the Pythagorean theorem to the right triangle shown in the diagram, we have $(s_1/2{)}^2+s_1^2=r^2$, which implies $s_1^2=\frac{4}{5}{r}^2$. Let $s_2$ be the side length of the square inscribed in the circle of radius $r$. Applying the Pythagorean theorem to the right triangle shown in the diagram, we have $(s_2/2{)}^2+(s_2/2{)}^2=r^2$, which implies $s_2^2=2r^2$. Therefore, the ratio of the areas of the two squares is $\frac{s_1^2}{s_2^2}=\frac{\frac{4}{5}{r}^2}{2r^2}=\boxed{\frac{2}{5}}$.