In some sense you could say that there are probably not as many sources as you might think: The conjecture of Kontsevich-Zagier says that when $\pi$ appears as some kind of period one should always be able to make some simple transformations of the involved integral to make it visibly equal to the standard definition. This of course is something number theorists/arithmetic algebraic geometers are used to do; try to find that two different $\pi$'s are the same (usually for some kind of motivic´´ reason).

This preamble is just to lead up to the amusing fact that wondering whether two different $\pi$ are the same or not is not the exclusive domain of arithmeticians. In his first book in the series of PDO's Hörmander proofs Fourier inversion formula by first proving that it is true up to a constant (using in effect the irreducibility of the Heisenberg representation of the Heisenberg group) and then uses two different methods for determining the constant (which of course involves $\pi$), one by computing the integral $\int_0^\infty e^{-x^2}$ and one by the Cauchy residue formula. He then writes ''The constants $2\pi$ in Cauchy's integral formula and in the inversion formula are therefore 'the same'...''