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?

One way to see it is if you consider a circle with radius $r$ and another circle with radius $r+\Delta r$ (where $\Delta r\ll r$) around the same point, and consider the area between the two circles.
As with any shape, the area is proportional to the square of a typical length; the radius is such a typical length. That is, a circle of radius $r$ has the area $Cr^2$ with some constant $C$. Now the area in between the two circles  has the area $\Delta A = C(r+\Delta r)^2-Cr^2\approx 2Cr\,\Delta r$. That relation gets exact as $\Delta r\to 0$.
On the other hand, the distance between the two circles is constant, and therefore for sufficiently small $\Delta r$ you can "unroll" this shape into a rectangle (again, the error you make when doing this vanishes in the limit $\Delta r\to 0$). That rectangle has as one side the circumference, $2\pi r$,  and as the other side $\Delta r$. Since the area of a rectangle is the product of its side lengths, we get as area $\Delta A = 2\pi r\,\Delta r$.
Comparing the two equations, we get $2Cr\,\Delta r=2\pi r\,\Delta r$, that is, $C=\pi$.
