Newton's Pi Approximation
This Demonstration gives Newton's approximation of based on calculating the area of a semicircle using an integral.[more]
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 .
Newton presented to 16 decimal places using 20 terms of the binomial series [1, p. 177].
 W. Dunham, Journey through Genius, New York: Penguin Books, 1990 pp. 174–177.