9711

Wagon Wheel Approximation of Pi

Let there be spokes of a wagon wheel. We only need to deal with the upper half of a unit circle. Divide the semicircle vertically into evenly spaced segments using bars.
The intersections of the spokes with the corresponding vertical bars creates truncated spokes of length , , rather than of length 1. Here and .
Notice that is undefined by an intersection because both the spoke and bar are vertical and coincide. The value of must be inferred from the pattern created by connecting the intersections—amazingly, approaches as increases!

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This Demonstration leads to one of the many beautiful expressions for :
,
(from the French mathematician Viète, 1593).
This formula comes from the cosine half-angle formula, which yields
,
,
, and so on.
Interpolation using rather than greatly improves the approximation.
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