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Why is the ratio of the circumference of a circle to its diameter independent of the circle?

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

What is the simplest way to prove that the ratio of diameter and circumference of any circle is the same number?

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marked as duplicate by Aryabhata, Sivaram, Chandru, Ross Millikan, Zev Chonoles May 31 '11 at 2:25

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The question should be refined to "In Eucleadian Geometry, What is the simplest way to prove that the ratio of diameter and circumference of any circle is the same number?" –  Arjang May 31 '11 at 3:21

1 Answer 1

Similarity.

Draw two concentric circles, and draw a regular n-gon in one of them. Connect the centre of the circles to the vertices of that n-gon by semilines the circle and label the intersection of those with the other circle. You get a similar n-gon.

Similarity shows that the ratios of the perimeters of the two n-gons is the ratio of the diameters of the circles; and the circumference is defined as the limit of the perimeter of n-gons.

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I'm a little unhappy with this method because of using the limit concept and also the assumption that the limit of perimeter of n-gons is equal to the circumference of circle. –  ashpool May 31 '11 at 2:25
    
@ashpool: Without limit $\pi$ doesn't exist –  user17762 May 31 '11 at 2:26
    
ashpool, as far as I know the circumference of the circle is defined as the limit of the polygons. Even in geometry, the standard argument that the circumference of the circle exists is the fact that the perimeter of inscribed polygons increases if the number of edges is doubled, while the perimeter of the circumscribed polygons decreases if the number of sides is doubled. Moreover their difference goes to zero.... This is BTW the way in which the ancient greeks found $\pi$..... –  N. S. May 31 '11 at 4:18

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