Are there two $\pi$s?

Solution 1:

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 $A = \frac{1}{2}Cr$.

Solution 2:

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.