Why Pi?

Why is Circumference/Diameter of any circle equal to pi?
Why is the relationship even linear?And why does the ratio come out to be a transcendental number?
Trying to understand this question,i decided to simulate the Buffon Needle experiment.The value is supposed to converge to pi as no of times experiment is conducted goes to infinity.However even after 5e+10 iterations,the computer was stuck at 1.86.Maybe its a fault of the rand() function in C.So i ttried better random generators which were there on the net.The value of 1.86 came earlier,but the computer still got stuck at it.
Well,the integral of 4/(1+(x*x))(pardon the poor formatting) within [0,1] also equals pi.So i posed the definite integral as a summation.This time 3.14159 came up soon enough.That was the limit of C double precision.Fortran could do better.

But i still can't help wondering ,why does such a simple ratio turn out to be a transcendental.