The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216 \pi$. What is the length $AB$?
$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 24$
Let the length $AB$ be $L$. Then, we see that the region is just the union of the cylinder with central axis $\overline{AB}$ and radius $3$ and the two hemispheres connected to each face of the cylinder (also with radius $3$). Thus the volume is
$9\pi L + \frac{4}{3}\pi(3)^3 = 9\pi L + 36\pi ( = 216\pi)$
$9\pi L = 180\pi$
$L = \boxed{(D)=\ 20}$