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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 

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   

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   
Associated Topics:
High School Constructions
High School Geometry
High School Triangles and Other Polygons

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