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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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blunders
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    Did you mean why is the ratio of circumference of a circle to diamter independent of the circle? – Aryabhata Feb 26 '11 at 19:08
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    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. – Ross Millikan Feb 26 '11 at 19:09
  • +1 @Moron: I would like to see the answer for your question. I had some doubts too but not a clear picture about it. Thanks – 0x0 Feb 26 '11 at 19:14
  • @Moron: Yes, that's correct -- updated the question, thanks! – blunders Feb 26 '11 at 19:19
  • @Sunil: Updated my question to Moron's wording, since he's meaning was the intent of my question. – blunders Feb 26 '11 at 19:21
  • @Ross Millikan: Updated the core question per Moron's suggestion, and removed the secondary question deal with any reasons why Pi might not be a constant. Have any additional feedback? Thanks! – blunders Feb 26 '11 at 19:25
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    I think the new version of the question is a good one. – Mike Spivey Feb 26 '11 at 19:28
  • @Mike Spivey: +1 Thanks for the edits to the question. – blunders Feb 26 '11 at 19:30
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    Duplicate: http://math.stackexchange.com/questions/23129/why-is-euclidean-geometry-scale-invariant – Qiaochu Yuan Feb 26 '11 at 19:38
  • @Qiaochu Yuan: +1 thanks, did search for a duplicate first, and figured there might be one - I'll take a look at the question you linked to. Cheers! – blunders Feb 26 '11 at 19:46
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    @blunders: Take a look at this also: http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits – Derek Jennings Feb 26 '11 at 19:47
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    @blunders: Here's another link: http://www.oocities.org/cf/ilanpi/pi-exists.html – Jonas Meyer Feb 26 '11 at 20:30

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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.

lhf
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    +1 Funny thing is, your answer makes sense to me -- the answer and question linked to as a duplicate... does not - oh, well. – blunders Feb 26 '11 at 19:50
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    @blunders: You may appreciate these answers (added again here as the identical link is a bit lost in the comments to the question): http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits – Derek Jennings Feb 26 '11 at 19:59
  • @Derek Jennings: +1 Thanks! – blunders Feb 26 '11 at 20:09