This note is an extension to my previous note. In that note, I asked the community to find the mistake in my solution yielding 0=1. In that note, this mistake was the multiplication of inifinty and 0 which is basically an indeterminate form. This note is a correct version of the previous note and my formula seems to give a good approximation for .
For any sided regular polygon inscribed in a circle of radius , our objective is to calculate the perimeter of the polygon.
The figure represents the given situation. is the center of the circle and , are edges of a regular polygon. and are perpendiculars drawn on line segments and respectively. Since perpendiculars from center bisect the chord, thus we have,
Therfore by RHS Criteration of congruency, .
Thus by CPCT, we have
For any regular polygon of sides, each angle is given by .
Therefore, . Thus .
Since it is a regular polygon,
But as approaches infinity the polygon tends to coincide the circle in which it is inscribed. Thus in that case the perimeter of the polygon becomes equal to the circumference of the circle in which it is inscribed.
Thus, Comparing the perimeter of the inifinite-sided polygon with the circumference of a circle we get,
Let us take a reasonable approximation for which resembles infinity. Thus, let
Plugging in the value of in the 2nd equation we get,
which yields us using a scientific calculator which is actually upto 30 decimal places!
Basically, if you want a better approximation just increase the power of 10.