What shape?

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 $3$ ).

$\displaystyle \begin{aligned} 3^2\pi L + \frac{4}{3}\pi(3)^3 & = 216\pi \\ 9\pi L + 36\pi & = 216\pi \\ 9\pi L & = 180\pi \\ L & = 20 \end{aligned}$