2023 AMC 10B Problem 20

Four congruent semicircles are drawn on the surface of a sphere with radius 2, as shown, creating a closed curve that divides the surface into two congruent regions. The length of the curve is \pi\sqrt{n}. What is n?

\textbf{(A)} 32 \qquad \textbf{(B)} 12 \qquad \textbf{(C)} 48 \qquad \textbf{(D)} 36 \qquad \textbf{(E)} 27

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.