Answer: B

Solution:

Let the square have side $s,$ so $s^2=40.$ Its base lies centered on the diameter, and a top corner at $\left(\tfrac{s}{2},s\right)$ lies on the circle.

Then $r^2=\left(\tfrac{s}{2}\right)^2+s^2=\tfrac{40}{4}+40=50.$ The semicircle area is $\tfrac12\pi r^2=\tfrac12\pi(50)=25\pi.$

Thus, the correct answer is B.