A straightforward approach is to calculate the area of the quarter circle in the first quadrant. Split the circle into 10 vertical segments, and use the trapezoid rule to get inside and outside estimates for the area. This requires nothing worse than square roots and slopes of tangent lines -- a minimum of analytic geometry. It's easy to see that bumping up the number of slices to 100 will reduce the error by a factor of around 10. This part can be programmed numerically. The next step is deeper: Writing out the binomial for sqrt(1-x^2), believing it converges (at least out to x=1/2), rearranging the terms to make an integral, and accepting integral(x^K) = x^(K+1)/(K+1), may take some selling. But after that, you actually have a reasonably convergent series for pi, related to arcsin(1/2) and the area of the portion of the unit circle between x=0 and x=1/2 and y>=0. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of Jason Holt Sent: Wednesday, July 12, 2006 2:05 PM To: math-fun Subject: [math-fun] Calculating pi 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun