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Proof that Pi is constant (the same for all circles), without using limits

How do we prove that the ratio of a circle's circumference to its diameter is a certain real number, the same for any circle (which we call pi)? Is there a pure geometric proof that the ratio is always the same?

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    [This question](http://math.stackexchange.com/questions/23129/why-is-euclidean-geometry-scale-invariant) is also relevant.2011-09-09

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One way to introduce lengths into Euclidean geometry is to use Rene Descartes' co-ordinate geometry. In this setup, using the methods of calculus, you can prove the said result using integration to compute the circumference(with an integral for the length of a curve).