The author wants feedback. Well, the article by Tong which is mentioned here https://math.stackexchange.com/a/2092943/124777 https://math.stackexchange.com/a/20

The author wants feedback. Well, the article by Tong which is mentioned here https://math.stackexchange.com/a/2092943/124777 https://math.stackexchange.com/a/2092943/124777 gives a deep and surprising reason for the relationship between surface area and volume. The referenced paper is currently available for download at https://www.maa.org/sites/default/files/0746834257453.di020775.02p0287y.pdf https://www.maa.org/sites/default/files/0746834257453.di0207... The same explanation is given by a video by Mu Prime Math on YouTube: https://youtu.be/cCwD38lw4ow https://youtu.be/cCwD38lw4ow IMO arguments involving the apothem of a polygon are obsoleted by Tong's insight. It becomes clear that it's just a coincidence that the "natural" parameterizations of these regular shapes give a neat answer. The basic observation that area scales as the square of a representative length, for any shape, comes into this discussion as well: The Pythagorean theorem: what is it about? Alexander Givental http://math.berkeley.edu/~giventh/papers/eu.pdf http://math.berkeley.edu/~giventh/papers/eu.pdf