Are there two $\pi$s? The mathematical constant $\pi$ occurs in the formula for the area of a circle, $A=\pi r^2$,
and in the formula for the circumference of a circle, $C= 2\pi r$. How does one prove that these constants are the same?

Here's something I like to call the "boundary rule":
$$
\begin{align}
& \phantom{={}} [\text{rate of motion of boundary}]\times[\text{size of boundary}] \\[10pt]
& = [\text{rate of change of size of bounded region}]
\end{align}
$$
Apply this to a growing circle.  The rate of motion of the boundary is $\dfrac{dr}{dt}$.  The size of the boundary is $C$.  The rate of change of size of the bounded region is $\dfrac{dA}{dt}$.
Since $A$ is the area and $r$ is a distance, we must have
$$A=(\text{some constant}\cdot r^2).\tag{1}$$
Hence
$$
\begin{align}
C\frac{dr}{dt} & = \frac{dA}{dt} \text{ by the boundary rule} \\[6pt]
& = \frac{dA}{dr}\cdot\frac{dr}{dt}.
\end{align}
$$
Canceling, we get
$$
C = \frac{dA}{dr}.
$$
Applying $(1)$ and differentiating gives you equality of the two constants.
Now the hard question: In what contexts would this qualify as a "proof"?
