A friend of mine recently asked me what pi is all about. This was rather remarkable, as he's failing out of high school, and people treat him like a failure. Turns out he's fascinated by physics and math. Anyway. So I described pi in terms of the experimental method of drawing circles and noticing that their circumference always seems a little more than 3 times their diameter. That led to a great discussion about experimental vs. theoretical science, and eventually he asked me how we went from experimental methods of calculating pi to computational approaches. I checked the wikipedia, but its approaches all seem to assume advanced math to prove that the approximations describe pi, and he hasn't gotten past algebra and geometry. At this point, the simplest approach I can think of is to write a program to choose random points in a circumscribed circle, then calculate the proportion of the points which fall within it. But that still has a heuristic element to it. Are there any plain arguments for the deterministic approximations? (For that matter, how about doing the same thing for the trig functions? I took the same approach for sin/cos/tan, talking about building a table of angle/ratio measurements, but I had no idea how to describe computational approaches.) -J