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Side Length of a Regular Octagon, Without Trigonometry

Date: 09/19/2003 at 17:12:28
From: Gail
Subject: Finding the length of the side of an octagon

I teach middle school math.  A friend asked me how to find the length
of one side of a regular octagon that has a (side-to-side) diameter of
16 feet.  I was not sure how to go about it, and would appreciate your

Date: 09/20/2003 at 02:02:06
From: Doctor Greenie
Subject: Re: Finding the length of the side of an octagon

Hi, Gail --

Think of the octagon as a square, with the four corners cut off at 
45-degree angles.  The pieces cut off are 45-45-90 right triangles.

Let's use "x" to denote the length of each of the legs of each of 
these triangles.  Then the hypotenuse of each of those triangles is 
x*sqrt(2).  But that hypotenuse is one of the sides of the octagon, 
so each side of the octagon has length x*sqrt(2)

You are calling the diameter of the octagon the distance from side 
to side; this is the same as the length of the side of the original 
square.  But if we picture the original square with the corner 
triangles marked but not yet cut off, then each side of that square 
(i.e., the diameter of the octagon) is made up of three segments in 
the ratio 1:sqrt(2):1.  The side of the octagon is the longer of 
these three pieces; then the fraction of the whole diameter of the 
octagon which is the length of one of the sides of the octagon is 
given by the ratio

  2 + sqrt(2)

So in general, if the diameter of the octagon is x, then the length 
of the side of the octagon is

  x * -----------
      2 + sqrt(2)

And in your particular case, with a given diameter of 16, the length 
of the side of the octagon is

  16 * -----------
       2 + sqrt(2)

I hope this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum 

Date: 09/26/2003 at 20:36:18
From: Gail
Subject: Thank you 

Dr. Greenie,

Thank you for your quick response.  I appreciate this very 
much and your explanation was understandable.  

Thank you again,
Associated Topics:
High School Euclidean/Plane Geometry

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