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How does one prove that a regular $n$-gon with perimeter $1$, approaches (becomes) a circle as $n$ goes to (or if $n$ is) infinity?

It is not enough to prove that all points become equal distance to origin, since this also holds for the limiting object of the graph of largest area drawn on the square graph and enclosed by a circle as we make the squares smaller and smaller.

Is the circle the only possible object with all points distance $r$ from another, and total perimeter $2\pi r$?

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    First, how do you *define* a limit of a sequence of polygons?2011-12-01
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    The unique object the n-gon's approaches?2011-12-01
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    You've seen how Archimedes went about this business, no?2011-12-01
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    Dear qqqqqqqqqqqqqqqquqqqqqqqqqqqqq, **please** change your user name to something less annoying!2011-12-01
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    It is not true that all point on the circle are at distance $r$ from another. Clearly there are points that lie very much closer together than $r$ on the circle. It is however true that all points are at a distance $r$ from a fixed point, the center. This is true only of the circle. (No need to consider the perimeter.)2013-01-15

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