[dnquark]

The original question has perfectly satisfying explanations accessible to anyone who knows algebra and is willing to spend a few weeks studying special relativity. cynicalkane above provides a great intuitive summary.

Likewise, generally speaking, when pi comes up in physics equations it is a geometric factor, either having something to do with the bona fide geometry of the system (e.g. the capacitance of a sphere) or with normalizing some relevant probability distributions. Name one example where we struggle to understand where pi comes from in physics formula!