For General Purposes
Pi (3.1416) can be approximated by the ratio 22/7. The ancient mathematicians similarly used 25/8 (Babylonia), 256/81 (Egypt), or 339/108 (India), each of which is within .02, better than most measurable tolerances of the period.
The ratio of the circumference of a circle to its diameter is the same for every circle, and that number is what we call π. In theory, if you take something perfectly round, measure the distance around it, then divide by its width, you could calculate π. However, most calculations of π are done using infinite series that depend on something called Machin-like formulas.
For example, according to an article on Wolfram's Mathworld, one such formula is: 1/4pi=183cot^(-1)(239)+32cot^(-1)(1023)-68cot^(-1)(5832)+12cot^(-1)(110443)-12cot^(-1)(4841182)-100cot^(-1)(6826318).
It is true that Pi (3.14..) is a ratio of circumference length to diameter length of any perfect circle. However, that statement doesn't explain how the value 3.14.. was first determined by Archimedes, so perhaps this explanation will be useful :
The basis for determining the ratio that is Pi (3.14..) rests on Archimedes' term "radian". Archimedes determined that at 57.296 degrees of arc, the "arc length" is equal to the "radius length" that is scribing the arc. Archimedes gave the term "radian" to this angle.
Simply dividing the radian value (57.296 degrees) into the circle value (360 degrees) produces ratio 6.28.. - Since the arc length at radian equals radius length, then 6.28.. times the radius length equals the circumference length.
To change the "radius/circumference" ratio to a "diameter/circumference" ratio, the ratio is simply halved to 3.14.., because the diameter is twice the radius.
It is really that simple. The ratio 3.14.. was given the mathematical language term of Pi by Archimedes.
The all around best formula for calculating pi involves every other number in the Fibonacci sequence. The Fibonacci sequence is a recursive alliteration. It starts out 1 1 2 3 5 8 13 21 34 55 89 144 233 377. Each term is the sum of the two preceding terms.
The formula is pi/4 = arc1/2 - arc1/5 + arc1/13 - arc1/34 + arc1/89 - arc1/233... "arc" here is abbreviated for the arc tangent of the number that it is attached to. The formula for calculating the arc tangent of numbers less than one is: arc't' = t^1/1 - t^3/3 + t^5/5 - t^7/7 + t^9/9 - t^11/11...
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