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Don has asked us how we would explain to a student why we do not correctly....
obtain the area of a sphere by adding the areas of what naively appear to
be infinitesimal triangles with vertices at a pole and infinitesimal bases
along the equator. This is a really good question as it concerns one of
the critical elements in setting up integrals which is, in turn, one of
the hallmarks abilities of a good physicist.
Clearly, as has already been discussed, the reason the "triangle method"
doesn't work is that the "triangles" aren't triangles. Nevertheless,
understanding this still leaves the original question unanswered.
At any rate, I still have no particularly good answer to the original
question, but I do hope to stimulate more consideration of this
interesting and, I think, important question.