43.
π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to
Archimedes,
[22] is to calculate the
perimeter,
Pn , of a
regular polygon with
n sides
circumscribed around a circle with diameter
d. Then
That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides
inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range: 3+10⁄71 < π < 3+1⁄7.
[23]
π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in
trigonometry and
calculus. However, some are quite simple, such as this form of the
Gregory-Leibniz series:
[24]
While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that nearly 300 terms are needed to calculate π correctly to 2 decimal places.
[25] However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let
and then define
then computing π10,10 will take similar computation time to computing 150 terms of the original series in a brute-force manner, and
, correct to 9 decimal places. This computation is an example of the
van Wijngaarden transformation.
[26]
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