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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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    (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.2011-12-02

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