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Physics
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Geometry

How is pi derived?

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March 05, 2014 6:11PM

Pi, of course, is the ratio of a circle's circumference to its diameter. This number is just slightly over three, and is irrational (it never repeats or terminates).

There are various formulas for deriving pi. You can even obtain an excellent approximation by dropping a pin on a floor of evenly spaced parallel lines. See the associated link for some of the formulas. Note: Machin's formula (using inverse tangents of unit fractions) is calculated in radians. In this respect it is, in effect, a definition of pi in terms of itself. Nevertheless it is a most curious relationship.

π/4 = 4tan-1(1/5) - tan-1(1/239)

More curious yet is the incredible relationship between pi and the natural log e:

eπi = -1, where i is the imaginary square root of -1. i2 = -1.