Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? $\textbf{(A) }25 \qquad$ $ \textbf{(B) }38 \qquad$ $ \textbf{(C) }50 \qquad$ $ \textbf{(D) }63 \qquad$ $ \textbf{(E) }75 \qquad$