Using A Chord To Find The
Circumference Of A Circle

E. F. Block
August 2015

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:

  1. 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
  2. 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
  3. 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)
  4. one then mutliplies this figure by the interation that yielded the very small angle in order to yield the approximation of the cicumference
  5. 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:

  1. 2 to the eleventh is 4096
  2. 1/2 of the diameter 11 times gives an angle of Theta = 0.0878906
  3. the sine of that angle is 0.00153398
  4. if the Radius = 1", then the chord is (1")(2)sin(0.0878906/2) = (1")(2)sin(0.0439453)
  5. or (1")(2)(0.00076699) = 0.00153398", and now multiply by 4096 gives 6.282854"
  6. 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

Front Page