One common definition of arclength is to just define it as a supremum of the set of lengths obtained by approximating your curve as a union of line segments. The natural analogue of this to the surface area of a surface in 3 space fails quite spectacularly thanks to constructions such as the Schwarz lantern, which shows we can approximate a cylinder by polyhedra whose surface areas approach infinity!
Is there an intuitive reason that polygonal approximation works so well for curves but fails so spectacularly for surfaces?
from Hot Weekly Questions - Mathematics Stack Exchange
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