Pi in Normal Distribution Equation

Does anyone know why the ratio of circumference to diameter is in the normal distribution equation? I know the normal distribution models most natural phenomena, but how the heck did Pi did up in there?

Thanks,
Dan

The same reason why it comes up in Buffon's needle problem, or Drunkard's walk problem - for that matter area of a circle problem.

In fact, the "complicated derivation" depends on the circumference of a unit circle being \(\displaystyle 2\pi\).