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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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marked as duplicate by Zev Chonoles Sep 9 '11 at 13:09

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
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
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This question is also relevant. – Zev Chonoles Sep 9 '11 at 13:09

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).

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