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Given Octagon Diameter, Find Side Length

Date: 04/09/2003 at 18:30:25
From: James
Subject: Octagonal Diameter vs Sides

If you have the diameter of an octagon, what formula gives you the 
length of the sides? I've found a formula where if you have the sides, 
you can find the diameter, but I don't know how to reverse the 
equation.

I have the short diameter or the side to side, but it would also be 
nice to know the vertex to vertex as well. I'm currently designing a 
house with several octagonal rooms. I appreciate all input you can 
give me. Thanks!

James Fields


Date: 04/11/2003 at 09:24:51
From: Doctor Ian
Subject: Re: Octagonal Diameter vs Sides

Hi James,

Draw a regular octagon, and make a mark at the center. From the 
center, draw a line to each vertex. You end up with 8 identical 
isosceles triangles.  

The angle at each vertex is 

  (8 - 2)*180
  ----------- = 135 degrees
      8

How do we know that? Pick any vertex, and draw diagonals to all but
the two adjacent vertices. You get 6 triangles. The sum of the 
interior angles of each triangle is 180 degrees, and the sum of the
interior angles of all the triangles is the sum of the interior angles
of the octagon, i.e., 6*180 degrees. In a regular octagon, the
interior angles are all equal, i.e., each one is (6*180)/8 degrees.  
You can use this strategy to find the interior angles of any regular
polygon.

So the base angle of each isosceles triangle is half that, or 67.5
degrees. 

Now pick one of the sides of the octagon, and draw a line from the
center of the octagon to the center of the side. You will have two
right triangles. Let's look at one of them:

        b
      ________
     /\t  |   \        t = 67.5 deg
    /  \  |    \
      c \ |a           a = (1/2)(diameter of octagon)
         \|

From trigonometry, we know that 

    tan(t) = a/b

         b = a/tan(t)

           = a/2.414 (approximately)

If you know the longer diameter, it's essentially the same story,
except that now you know c instead of a, and

    cos(t) = b/c

         b = c cos(t)

           = c * 0.383 (approximately)

Does this help?

By the way, that sounds like it's going to be an interesting house!

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

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