The ratio of the volumes of Cone:Hemisphere:Cylinder is 1:2:3.
Make each with a radius of r and and height of r. (The formulas work out nicely. This is left to the reader as a nice exercise 😉
Here’s a nice gif of the concept (click to see the animation):
This was found on a very nice page: http://blog.zacharyabel.com/tag/cavalieris-principle/
See tweet with video: https://twitter.com/74WTungsteno/status/1345852691948384259?s=20
Also on the same page is the
Archimedes’ Hat-Box Theorem: If we draw any two horizontal planes as shown below, then the portions of the sphere and the cylinder between the two planes have the same surface area.