Rhombus $ABCD$, with side length $8$, is rolled to form a cylinder of volume $24$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$?

1. $2\pi r=8$ (side becomes circumference) and $\pi r^2h=24$.
2. $r=\frac4\pi$, so $h=\dfrac{24}{\pi r^2}=\dfrac{24\pi}{16}=\dfrac{3\pi}2$.
3. $\sin\angle ABC=\dfrac h8=\boxed{\dfrac{3\pi}{16}}$.