What do you do about 4πr²?

Extract

The mensuration formulae for curved objects ought to be one of the highlights of everyone’s elementary mathematical education. I get my pupils at 13 and they've already been taught that the area of a circle is πr². It is very upsetting to find that they usually have seen no sort of justification of this result, so they don’t think of it as surprising when π, which was introduced in the circumference formula, mysteriously pops up again in the area formula.

This year I thought I would put some effort into justifying the surface area formula for the sphere. I don’t claim that what follows is very original, and of course the main idea goes back to Archimedes, but I hope this article will save readers a bit of work at least.