Answer: C

Small Hint:

Let the circle have radius $R$ and the semicircle-square have side $s$

Big Hint:

For the semicircle-square, a top vertex gives $(\frac{s}{2})^2+s^2=R^2$

Solution:

For the square in the semicircle, put its base on the diameter. A top vertex has horizontal distance $\frac{s}{2}$ from the center and vertical distance $s,$ so $$\left(\frac s2\right)^2+s^2=R^2,$$ giving $s^2=\frac{4R^2}{5}.$ A square inscribed in the full circle has diagonal $2R,$ hence area $2R^2.$ The ratio is $$\frac{\frac{4R^2}{5}}{2R^2}=\frac25.$$

Thus, the correct answer is C.