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

How can you prove that the ratio of the circumference to the radius is a constant (regardless of what the constant is) using elementary geometry. Thank you

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hargun3045
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    If you compute that ratio for one circle, just move your head closer or farther away from the same circle. You obviously don't change the ratio when you do that. I don't know if that "proof" is sufficiently rigorous. – Dimitrije Kostic Dec 02 '11 at 21:09
  • @hargun3045 - Although use calculus is more rigorous, i will give you the link(look at rearragement proof section): https://en.wikipedia.org/wiki/Circle_area – Victor Dec 02 '11 at 21:10
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    Related post: http://math.stackexchange.com/questions/3198/. – Srivatsan Dec 02 '11 at 21:16
  • More importantly, I want to show that the circumference is proportional to the radius. Maybe someone should edit the post and avoid the discussion of pie at the moment. I'll look up at the links though. – hargun3045 Dec 02 '11 at 21:21
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    @hargun3045 I recommend T..'s answer in the link I provided. It asks a very similar question as yours. – Srivatsan Dec 02 '11 at 21:33
  • Ok. sorry for repetitiveness. I'll read that and come back here to ask if I've not understood. ( So far, I've never understood!) – hargun3045 Dec 02 '11 at 21:35
  • I explicitly showed this in this post http://math.stackexchange.com/questions/85217/why-is-this-series-of-square-root-of-twos-equal-pi/85219#85219 – David Mitra Dec 02 '11 at 21:51
  • @David, How does your answer explain how the circumference is proportional to the radius? – Srivatsan Dec 02 '11 at 21:57
  • @Srivatsan In each approximation, $P_n= b_n r$. The circumference of the circle is then $Cr$. where $C=\lim b_n$. – David Mitra Dec 02 '11 at 22:30
  • (1) Draw the inscribed and circumscribed regular $n$-gon for each circle. (2) Show that in each case the ratio of perimeter to radius is independent of radius. (3) Hand wave and say that as $n\to\infty$ the perimeters of the inscribed and circumscribed $n$-gons approach the perimeter of the circle. (4) QED. The weak point is (3). There is a need to define carefully the meaning of perimeter of a curve, and of course we have not done that. – André Nicolas Dec 02 '11 at 22:32

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