A couple of days ago I tried to self study the mathematical  explanation of pi (?) as well as ended up with very interesting results as well as ideas  
First of all lets answer the classic question; What is pi?
Pi or ? is a mathematical constant whose value is the ratio of any circle’s circumference to its diameter in Euclidean space… (Wiki)
From its definition it is pretty easy to find it subject to another mathematical function: sinus.
I can hear that two questions rising up in minds rapidly;
1) Why would I require that? -I don’t know the answer, just curiosity  
2) How? - Answer is in the rest of the post… continue… cmon, go on!
Since it is hard to write mathematical formulas as well as drawing shapes on our stupid HCI (human computer interaction; keyboard, mouse, etc.) slavery, I prefered to write them on a notebook as well as took their shots on behalf of the ease of understanding as well as publishing.
Relation Between Pi as well as Sinus Function


At first galance it is necessery to know that a circle can be expressed by infinite number of triangles,

so let’s start with the an estimated all basic equilateral; square.




The first half of the image above shows how to find the area of the square by using its half-diagonal which is the first step to do inductive reasoning.

And the second half is a generalisation of the formula on behalf of an equilateral with ‘n’ sides/corners.
As much as we increase the number of sides of the equilateral as much as it approximates to a circle,

Let’s think that we have an equilateral with infinite sides, then it turns into a circle which means in the result formula if we give a very big value to ‘n’ as well as 1 to ‘r’, then the result approximates to pi.


instead of subject to the number of sides in the equilateral, we desire to be dependent to the angle alpha, to do that we simply replace n with 360/alpha.


the conclusion of this part is awesome; we can find the value of sinus in the degree range of zero as well as ten, with an acceptable error (+0.001) by only one multiplication.
Error Rate
….
to be continue
….