A Spinning Top

A part of a solid can need its own chase before it gives a volume. A cone given by its slant height needs its height first, wherever it sits.

This rocket is a cone on a cylinder, both of radius 6, and the cone is given by its slant height 10. Its height comes first: h^2 = 10^2 - 6^2 = 64, so h = 8. Then the cylinder holds pi × 6^2 × 3 = 108×pi and the cone 1/3 × pi × 6^2 × 8 = 96×pi. Together: 108×pi + 96×pi = 204×pi.

Plan before you tap: list what each part needs, find what is missing, then the volumes.