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Cone, Hemisphere, Cylinder Relationship


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/

Surface Area

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.

See also: https://www.mathsisfun.com/geometry/cone-sphere-cylinder.html