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?
 A: The formula $C = 2\pi r$ is the definition of $\pi$.  That means when people ask what $\pi$ is, the answer is $\frac{C}{2r}$.
So the real question here is why is the area of a circle $\frac{1}{2}Cr$?  For an intuitive answer imagine cutting a circle into pizza slices and stacking then as in this picture:
$\hspace{5.5cm}$

If your pizza slices are thin enough then that shape is almost a rectangle and we can get it's area by length times width.  The width is the radius and the length is half the circumference.  Thus $A = \frac{1}{2}Cr$.
A: I believe Archimedes argued that as far as area is concerned, a circle is equivalent to a triangle with the circumference as a base, and the radius as altitude on that base.
A: 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, $C=\pi$.
A: 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"?
