2007 AMC 12B Problem 19

Let $\theta=\angle ABC.$ The base circle has circumference $6,$ so its radius is $\dfrac{6}{2\pi}=\dfrac3\pi.$ The height of the cylinder is the rhombus altitude $6\sin\theta.$ The volume is $$\pi\left(\dfrac3\pi\right)^2(6\sin\theta)=\dfrac{54}{\pi}\sin\theta=6,$$ so $\sin\theta=\dfrac{\pi}{9}.$ Thus, the correct answer is A.