Pi and PolygonsDate: 03/14/99 From: Anonymous Subject: Pi and Polygons A simple iterative derivation inscribes an n-polygon inside a circle until the limit approaches pi (the circumference of the unit circle). Is there a formula to find the angle of an n-sided polygon given that it has x sides? This formula is obviously convergent. Take the limit as theta -> 180 (or pi radians) and the infinity*0 = pi.? Date: 03/14/99 From: Doctor Ken Subject: Re: Pi and Polygons This method of finding successive approximations to Pi is one of the oldest methods known, because it is one of the easiest to understand, and you can draw nice pictures to explain it. If a regular polygon has n sides, then we can draw lines from its center to all the vertices, and these lines will divide the pie-shaped picture into n wedges. Each wedge's central angle will have a measure of 360/n degrees. So, we can draw this picture of one wedge, where C is at the center of the polygon: A /| / | / | / | / | C /__________| D \ | \ | \ | \ | \ | \| B Segment AB here is one side of the original regular polygon. Since angle ACB is 360/n, angle ACD is 180/n. Therefore, if the length of AC is 1/2, the length of AD is sin(180/n)/2. Therefore, the length of AB is sin(180/n), and the perimeter of the entire polygon is n*sin(180/n). So, you are correct: when you let n go toward infinity, sin(180/n) will tend towards zero. Since we know that a circle whose radius is 1/2 has a circumference of Pi, the 0 and infinity balance each other in the limit. Of course, there is a practical problem with all of this. In order to calculate the sines, you need to know a thing or two about Pi. It is true that for some special angles like 30, 45, and 60 degrees (and their sums and differences, etc.) you can write down an explicit elementary expression for their sines, this is not true of some other angles like 180/11. So, how do we calculate the perimeters without knowing Pi already? We need to find some trick, or we need to find some other method entirely of approximating Pi. And that is where a great part of the glorious history of mathematics starts. For more about Pi, its role in history, and the various attempts to know it better, check out the excellent book _A History of Pi_ by Petr Beckmann. People have done some pretty clever things to get to know Pi. - Doctor Ken, The Math Forum http://mathforum.org/dr.math/ |
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