for example, just about every interesting plain curve involves π. See: Visual Dictionary of Special Plane Curves.
the only reason i can think of is that it's the circular function. (that is, the essence of trig function. (See: Sine Curve.))
but it happens in probability too. For example: Buffon's needle.
the Wikipedia article indicates that Buffon's needle is part of a problem in in geometric probability. So, perhaps that ties to the circle, therefore π. Circle itself can be considered as a result of circular function. i.e. the trace of a constant length with one end fixed and the other end moves in a constant speed. So, we can consider that the relation of Buffon's Needle is tied to circular function.
but are there other occurrence of π that has little to do with circle? (For example, it appears in series often. How's this explained?)