How can we prove that the circumference of a circle is proportional to its diameter?

The parametric equation of a circle of radius `r` is given by

`x=rcos(t``)`

`y=rsin(t)`

where `t=[0,Omega] ` and `Omega ` is the total radians or degrees in a circle. We assume that given any circle the total degrees is always the same, therefore `Omega` is constant.

To measure the length of a curve, in particular the length of the circumference of our circle, we use the following formula:

`C=int_0^Omega sqrt((dx/dt)^2 + (dy/dt)^2) dt`

The derivatives `dx/dt` and `dy/dt` are simple to evaluate. We have `dx/dt=-rsin(t) ` and `dy/dt=rcos(t) ` . Plugging in the derivatives into the integral,

`C=int_0^Omega sqrt((-rsin(t))^2 + (rcos(t))^2) dt`

`= int_0^Omega sqrt(r^2sin^2(t) + r^2cos^2(t)) dt `

By the pythagorean theorem `sin^2(t)+cos^2(t)=1`. Therefore

`C= int_0^Omega sqrt(r^2) dt = int_0^Omega r dt = r int_0^Omega 1 dt = r [Omega-0] = r Omega` 

This shows that `C=r Omega=2r Omega/2`, the ratio between the circumference and the diameter of the circle is `Omega/2` regardless the size of the circle.