I'm trying to correlate $\pi$ to the circumference and diameter of a unit circle. Consider a unit circle and represent it in terms of an n-gon. Join each vertex to its center. The angle subtended is $\theta^\circ = \frac{360^\circ}{n}$. Now the base of each of these triangles (formed by the radii and the chord on the arc) is of measure $2\sin\left(\frac{\theta^\circ}{2}\right)$ which can be computed by the cosine rule. The circumference is therefore supposed to be $\displaystyle\lim_{\theta^\circ \to 0^\circ} \left\{n\cdot2\sin\left(\frac{\theta^\circ}{2}\right)\right\} = \displaystyle\lim_{\theta^\circ \to 0^\circ} \left\{\frac{360^\circ}{\theta^{\circ}}\cdot2\sin\left(\frac{\theta^\circ}{2}\right)\right\} = 360^\circ \cdot \displaystyle\lim_{\theta^\circ \to 0^\circ} \left\{\frac{\sin{\left(\frac{\theta^{\circ}}{2}\right)}}{\left(\frac{\theta^{\circ}}{2}\right)}\right\} = 360^{\circ}$
So does $360^{\circ} = 2\pi$? Isn't that supposed to be $2\pi \text{ rad}$ instead of just $2\pi$? We also know that $\pi = 3.14...$ So does $180^\circ = 3.14...$?
I think what you are getting confused is with degree and radians. Now, as we know angles can be expressed in degree as well as radians. When in any equations we use angles in degree what we get is degree related value. Whereas radians outputs only number. There is no unit to attach with the answer. Lets take an example:-
$f(x) = \frac {\sin x}{x}$ , for $x \in \mathbb R$ thereby $f(x)\in\mathbb R$
Now if we take $x = 30^\circ $. The answer is $f(30^\circ)=0.0167\ degree^-1$
Now if we take $x=30\ radians\ or\ only\ 30$. The answer is $f(30)=-0.988$
The reason behind it is we assume radians as unit less, calculations using radians give pure numbers. But with degree we have to associate it with the answer. Another prominent Example would be Angular Velocity.
$V = \dot \theta = \frac{d\theta}{dt} $
If we used $degrees\ as\ unit\ of\ \theta,\ then\ the\ unit\ of\ V\ is\ degree.s^-1, But\ if\ radians\ is\ used\ then\ radians.s^-1\ or \ simply\ s^-1 $
And also the radians is defined as:
$1\ radians$ is defined as the angles subtended by an arc such that the arc length is same as the radius.
As we can see its a ratio, an unit less substance.