Circle of Life of PI

As I was thinking about PI, I felt may be I could compute its value from a limiting case.

Could PI could be defined as the ratio of perimeter of a "cyclic equilateral polygon" to its "circum-diameter" under the limits of inifinite number of sides for this polygon?

Let me explain.

We know that pi is a irrational constant defined as the ratio of circumference of a circle to its diameter. 

i.e.  PI = C / d 

Next, we are familiar with a 'cyclic quadrilateral' which is any quadrilateral that can be circumscribed i.e. all its vertices lie on a circumference of a circle. Finally, equilateral is referred to any polygon with sides of equal length.

Thus, "cyclic equilateral polygon" is any polygon with (n) sides of equal length that can be circumscribed within a circle. The diameter of such a circumscribing circle is termed circum-diameter.

In the limiting case of the number of sides of a polygon tending to infinity, we can imagine that a "cyclic equilateral polygon" will become a circle. Then the perimeter of a "cyclic equilateral polygon" under such the limits becomes the circumference (of the circle).

Let r be the circum-radius. Let us consider a triangle formed by two radii and one side of this polygon. Let theta be the included angle. Now, let us try to compute the length of the segment (denoted 's') of this "cyclic equilateral polygon".

Using acute angle theorem, we can show that 

s^2 = r^2 + r^2 - 2* r*r*cos(theta)

s^2 = 2*r^2 - 2* r^2*cos(theta)

s^2 = 2*r^2 * [1 - cos(theta)]

s^2 = 2*r^2 * [2 * sin^2(theta/2)]

s^2 = 4 * r^2 * [sin(theta/2)]^2

s = 2 * r * sin(theta/2) ....... equation A

On the value of theta, the included angle, we see that entire 360 degrees is divided into n parts one for each triangle. Thus, we see that the value of theta is 360/n. 

Substituting this in the equation A, we get:

x = 2 * r * sin((360 degrees/n)/2) = 2 * r * sin(180 degrees/n)

This allows us define the perimeter (P) of the "cyclic equilateral polygon" as 

P = n * s = 2 * n * r * sin(180 degrees/n)

Now, PI = lim (n -> infinity) {P / d} = lim (n -> infinity) {2 * n * r * sin(180 degrees/n) / d}

But d = 2*r, thus we get

lim (n -> infinity) {n * sin(180 degrees/n) }

Sine function is defined only for arguments in radians. i.e., 180 degrees is PI radians. 

We know that limit (x->0) {sin(x)/x} = 1

Thus we get,

lim (n -> infinity) {n * sin(PI/n) } = PI 

This is circular logic..! I can not compute the value of PI assuming that I know its value! (Here PI was used to define the angle.)

Interestingly, we can calculate PI from random numbers! It may seem unbelievable, but read this article for an explanation on how this can be achieved. The author points to an article that proposes to employ this technique for spam control!