# How to prove that Pi exists? [duplicate]

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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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Hand waving answer: it is a ratio, and ratios are constant with scaling. Making a bigger or smaller circle is just scaling, so the ratio stays the same. –  user1729 Sep 9 '11 at 13:07
Is a simple scaling argument not acceptable? –  Zhen Lin Sep 9 '11 at 13:07
This question is also relevant. –  Zev Chonoles Sep 9 '11 at 13:09