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Newton's Pi Approximation

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This Demonstration gives Newton's approximation of based on calculating the area of a semicircle using an integral.

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The area under the arc (light blue) is equal to the area of the sector minus the area of triangle (light green), that is, .

On the other hand, the semicircle has equation , so . Set and .

Therefore, .

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Contributed by: Izidor Hafner (August 2013)
Open content licensed under CC BY-NC-SA

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Details

Newton presented to 16 decimal places using 20 terms of the binomial series [1, p. 177].

Reference

[1] W. Dunham, Journey through Genius, New York: Penguin Books, 1990 pp. 174–177.

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