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This question already has an answer here:

Can anyone explain what's wrong with this?

enter image description here

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marked as duplicate by Henry Swanson, Hakim, 5xum, user91500, Git Gud Jun 2 '14 at 11:55

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The problem is that you think that if a sequence of curves $\gamma_1,\gamma_2,\dots$ approaches the curve $\gamma$, the lenghts of $\gamma_i$ should converge to the length of $\gamma$. This is not true.

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    $\begingroup$ Also, it's got nothing to do with $\pi$ per se. You could use the same reasoning to argue that the distance from (0, 0) to (1, 0) is $\sqrt{2}$. $\endgroup$ – StumpyLeg Jun 2 '14 at 19:12

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