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The more general question is: If you approximate a curve by a series of straight lines, how do you know when you use smaller and smaller straight lines, in the limit, the sum of lenghs of these small straight lines will be the length of the curve?
I suspect it has something to do with differentiability. If you approximate a curve by smaller and smaller corners, in the limit, the 1st order differential of the resulting curve is no where continuous. Whereas if you aproximate the curve by the hypotenuse of the small corners, the resulting curve's 1st differential will be continuous when the hypotenuses become smaller and smaller; and the lenghts of the hypotenuses will be able to approximate the length of the curve.
I will leave you to fill in the missing details on why a continuous 1st order defferential will enable the lengths of a series of small straight lines to be able to approximate the length of the curve.
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