If a cube can be construed as a series of squares stacked one upon the other, can a sphere be considered as a series of circles stacked one upon the other?

Question

If a cube axbxc can be said that it is a series of “c” squares axb stacked one top of another, then volume of cube is sum of areas of all the c squares

ab+ab+…. ab c times, so abc.

Similarly could a sphere of radius r can be seen as a series of circles stacked one over other each with increasing radius from 0 to r for the top and bottom halves of sphere independently.

In that case volume of sphere is the twice the sum of all the areas of those circles.

2*pi*[ r^2+(r-1)^2+……0)

In: 0

Answer 1

The thing with cubes is that the cut section at every height has the same size and shape, allowing the “squares” to have a height – i.e. be 3-dimensional with height 1 each so you only have to add c of those “squares”.

This doesn’t work with spheres, because if you made the circles 3-dimensional before stacking them then the result would no longer be a sphere. So you’d have to keep the circles 2-dimensional, without height.

A way to resolve this is to still treat those circles as 3-dimensional but with infinitesimal (infinitely small) height, but that’d still require you to add up infinitely many of them to get the whole sphere. This would essentially compute the volume of the sphere as an integral but it’s much more complex than the shortcut you can use with cubes.