Pi () is the ratio of a circle's circumference to its diameter. It is thus a fundamental constant of the universe, and yet, the specific number we use is an arbitrary choice. It would, for instance, be just as valid if we'd defined to be the ratio of a circle's circumference to its radius. In this case, all the formulae we have now would still work, except we'd replace everywhere we currently have with /2 and everywhere we have 2 with .
I wonder if that might not have made more sense. Certainly, that the circumference of the unit circle is 2 means that 2 appears in a lot of equations. On the other hand, 2 is easier to work with and write than /2, so exchanging a lot of 2 for would only be an advantage if we didn't have enough changes of to /2 to more than make up for it.
Of course we are stuck with being what it is for the simple reason that diameters are easier to measure than radii when you're working with things like tables and trees. Changing it would be crazy impossible; there may be no more firmly rooted number in all of math.