16-sided Regular Polygon
Date: 07/31/2001 at 18:18:06 From: Tiffany Travis Subject: 16 sided regular polygon I have tried to construct a 16-sided polygon, but my attempts have been unsuccessful, and I don't know how to construct one. If you could help I would appreciate it.
Date: 08/01/2001 at 13:43:15 From: Doctor Jubal Subject: Re: 16 sided regular polygon Hi Tiffany, Thanks for writing to Dr. Math. There's a straightforward way to construct any regular polygon whose number of sides is a power of two (square, octagon, 16-gon, etc...) We'll start with a pair of perpendicular lines. Let's call the point where they intersect point P. If you draw a circle centered at P, the four points where that circle intersects the two perpendicular lines are the corners of a square. The two perpendicular lines form four 90-degree angles at P. If you bisect all those angles, you'll have eight 45-degree angles at P, and the eight points where the circle centered at P intersects the original lines and the angle bisectors are the vertices of a regular octagon. You can repeat the process, bisecting the 45-degree angles to make sixteen 22.5-degree angles, and the sixteen points where the circle at P intersects the two original lines or any of the angle bisectors are the vertices of a regular 16-gon. You can make 32-gons, 64-gons, or any regular 2^n-gon in the same fashion. All you need is a consruction to bisect an angle. I searched the Dr. Math archives using the keywords construct angle bisector (all, any order) and found the page Geometry Constructions with Compass and Straightedge http://mathforum.org/dr.math/problems/zaidi11.13.98.html It contains instructions for bisecting an angle with compass and straightedge, complete with illustrations. Using that method, you should be able to construct any regular 2^n-gon you like. As a generalization, we started with a square inscribed in a circle and kept bisecting the angles at the center of the circle to make a regular octagon, 16-gon, etc. If you start with an equilateral triangle inscribed in a circle, you can use the same method to construct a regular hexagon, dodecagon, 24-gon, etc., and if you start with a pentagon inscribed in a circle, you can use it to construct a regular decagon, 20-gon, 40-gon, etc. Using this method, if you can construct a regular n-gon, you can construct a regular (2^k)*n-gon. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Jubal, The Math Forum http://mathforum.org/dr.math/
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