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Date: 30-7-2020
1480
Date: 14-3-2020
933
Date: 26-8-2020
2019
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Let and be the perimeters of the circumscribed and inscribed -gon and and the perimeters of the circumscribed and inscribed -gon. Then
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The first follows from the fact that side lengths of the polygons on a circle of radius are
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so
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But
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Using the identity
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then gives
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The second follows from
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Using the identity
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gives
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Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi ().
REFERENCES:
Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 186, 1965.
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