Let and be the perimeters of the circumscribed and inscribed -gon and and the perimeters of the circumscribed and inscribed -gon. Then
(1)
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(2)
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The first follows from the fact that side lengths of the polygons on a circle of radius are
(3)
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(4)
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so
(5)
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(6)
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But
(7)
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(8)
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Using the identity
(9)
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then gives
(10)
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The second follows from
(11)
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Using the identity
(12)
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gives
(13)
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(14)
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(15)
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(16)
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Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi ().