Just find a nice demonstration for volumes and area of the n-spheres here.
First he demonstrates the formula for a general pyramid (by dividing a general cube in pyramids) to get the Volume from the Area. The case for the 3 dimensional sphere he settles by comparing its area with the cylinder, actually he just claims it is possible, pointing to a more famous argument that I already saw before here for example. Then he claims that this kind of argument always works, that I didn't knew!! And derives the general formula from it. It is actually a recurrence formula but we can certainly find the general formula from that.
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