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Possible Duplicates:
Why is Euclidean geometry scale-invariant?

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

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    $\begingroup$ Did you mean why is the ratio of circumference of a circle to diamter independent of the circle? $\endgroup$
    – Aryabhata
    Feb 26, 2011 at 19:08
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    $\begingroup$ What definition of $\pi$ are you using that might make it not a constant? What would cause it to vary? Vote to close as not a real question. $\endgroup$ Feb 26, 2011 at 19:09
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    $\begingroup$ I think the new version of the question is a good one. $\endgroup$ Feb 26, 2011 at 19:28
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    $\begingroup$ Duplicate: math.stackexchange.com/questions/23129/… $\endgroup$ Feb 26, 2011 at 19:38
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    $\begingroup$ @blunders: Take a look at this also: math.stackexchange.com/questions/3198/… $\endgroup$ Feb 26, 2011 at 19:47

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Because when you change the scale, both the diameter and the circumference change by multiplying by the same scale factor, since they are both of dimension 1. Thus, their ratio is independent of the scale factor. Since all circles are similar, the ratio is the same for all circles.

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    $\begingroup$ +1 Funny thing is, your answer makes sense to me -- the answer and question linked to as a duplicate... does not - oh, well. $\endgroup$
    – blunders
    Feb 26, 2011 at 19:50
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    $\begingroup$ @blunders: You may appreciate these answers (added again here as the identical link is a bit lost in the comments to the question): math.stackexchange.com/questions/3198/… $\endgroup$ Feb 26, 2011 at 19:59
  • $\begingroup$ @Derek Jennings: +1 Thanks! $\endgroup$
    – blunders
    Feb 26, 2011 at 20:09

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