Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$?
$\mathrm{(A)}\ \frac{\pi}{9} \qquad \mathrm{(B)}\ \frac{1}{2} \qquad \mathrm{(C)}\ \frac{\pi}{6} \qquad \mathrm{(D)}\ \frac{\pi}{4} \qquad \mathrm{(E)}\ \frac{\sqrt{3}}{2}$
$V_{\mathrm{Cylinder}} = \pi r^2 h$
Where $C = 2\pi r = 6$ and $h=6\sin\theta$
$r = \frac{3}{\pi}$
$V = \pi \left(\frac{3}{\pi}\right)^2\cdot 6\sin\theta$
$6 = \frac{9}{\pi} \cdot 6\sin\theta$
$\sin\theta = \frac{\pi}{9} \Rightarrow \mathrm{(A)}$