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This article details the use of chords to determine the approximation of the cicumference of a circle without using the formula, C = (2)Pi(Radius).
The rational is as follows:
- if one chooses an angle of sufficient character, in this case very small, one may add up all of the chords of that angle within the circle to get a very good approximation of the circumference of that circle
- starting with the diameter, that is a chord of special case, one halves the successive areas within the circle to "n" iterations, or 2 to the power of "n", until the angle of the "n"th generation is very small
- one then determines the sine of that very small angle in order to find, the chord of Theta using the formula, crd Theta = 2 sin (Theta/2) and chord = Radius (crd Theta)
- one then mutliplies this figure by the interation that yielded the very small angle in order to yield the approximation of the cicumference
- the number of the iteration will determine the closeness of the value received to the value given by using the formula C = (2)Pi(Radius), according to the significent figures desired
An Example:
- 2 to the eleventh is 4096
- 1/2 of the diameter 11 times gives an angle of Theta = 0.0878906
- the sine of that angle is 0.00153398
- if the Radius = 1", then the chord is (1")(2)sin(0.0878906/2) = (1")(2)sin(0.0439453)
- or (1")(2)(0.00076699) = 0.00153398", and now multiply by 4096 gives 6.282854"
- and C = (2)Pi(Radius) = (2)(3.141592)(1") = 6.283184", which shows that our approximation is indeed very close to the value of the circumference formula
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