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The total base area for the whole shape is held constant by the requirements of your question,

This proof is in 3D, 2D is much easier, but uses the same logic.

Sorry but this didnt help much. How is this cube related to a circle or a sphere?

It is the starting point: as you increace the number of faces, the volume goes up

http://www.math.utah.edu/~treiberg/isoperim/isop.pdf

thanks