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Date: 8-10-2020
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Date: 10-3-2020
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Date: 2-3-2020
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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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(1) |
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(2) |
The first follows from the fact that side lengths of the polygons on a circle of radius are
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(3) |
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(4) |
so
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(5) |
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(6) |
But
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(7) |
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(8) |
Using the identity
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(9) |
then gives
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(10) |
The second follows from
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(11) |
Using the identity
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(12) |
gives
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(13) |
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(14) |
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(15) |
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(16) |
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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