Pi typically shows up when whatever you’re looking at has a relation to circles or spheres. For example, electromagnetism formulae contain pi because, among other reasons, abstract point charges have a spherically symmetric electric field. Pi also shows up when there’s a relationship to e. Recall the Euler identity that Chronos pointed out, particularly the general form e^(i*y) = cos(y) + i*sin(y). This arises from the Taylor series of the functions e^x, cos(x) and sin(x).

http://home.ecn.ab.ca/~jsavard/math/ide01.htm

e^(i pi) = -1 is just a specific case of this.

e is so ubiquitous, and is so simply related to pi, that pi just happens to occur in a lot of real-world phenomena*.

So, this raises the question, Why is e all over the place? I really don’t know. It has some very simple definitions, so I guess it’s pretty much bound to show up in some places. When I’ve done physics problems, the formulae we derived typically involved inadvertantly taking derivatives of 1/x, yielding the natural log of x.

Wherever e shows up, you can probably expect pi to follow. e plays a prominent role in growth and decay, compound interest, normal and gamma probability distributions**, the distribution of prime numbers, catenaries (the shapes wires assumes when suspended from both ends), blah blah blah

*The fact that a formula with ‘imaginary’ numbers has any relation to the real world may be puzzling, but actually there is nothing imaginary about imaginary numbers. ‘Real’ and ‘imaginary’ are misnomers. Many physical phenomenon require more numbers to describe than we have on the real number line. The solution is to take a second number line, called imaginary, and orient it perpendicular to the real number line so that the two intersect at the origin so that we now have an plane of numbers rather than a line.

**Since these were mentioned earlier: the normal distribution is a bell-shaped curve given by 1/Sqrt(2 pi) * e^(-z^2/2). The pi-containing factor appears because the area under f(z) = e^(-z^2/2) is equal to Sqrt(2 pi), and we intuitively want the total probability of something to equal one.