A cylinder has radius 3 and height 8. A cone with radius 6 has the same volume as the cylinder. What is the height of the cone?

Find the cylinder's volume: $V = \pi r^2 h = \pi (3)^2 (8) = 72\pi$. The cone has volume $\frac{1}{3}\pi r^2 h$ with radius 6, so $\frac{1}{3}\pi (36) h = 12\pi h$. Setting $12\pi h = 72\pi$ gives $h = 6$. A common wrong answer is 2, which comes from forgetting the factor $\frac{1}{3}$ in the cone's formula (solving $36h = 72$). Another is 8, assuming the heights are equal, and a third is 18, which results from using radius 3 for the cone while keeping the factor $\frac{1}{3}$. The cone formula needs the $\frac{1}{3}$ and the cone's own radius.