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
