Octagon Side Lengths

Date: 08/22/2001 at 22:26:26
From: Rob
Subject: Octagon

I need the formula to compute the length of each side of a octagon. I 
only know the dimension of the octagon from one side to the other, 
which is 8 feet. (It is 8 ft. from the center of one side to the 
opposite side.) How can I find the lengths of the 8 sides? 

I tried to draw it out on graph paper starting with an 8-foot sguare 
and scaling it with my rule using 22.5-degree angles, but I was not 
sure about my accuracy.

Date: 08/24/2001 at 08:58:13
From: Doctor Paul
Subject: Re: Octagon

We can solve your problem using a bit of high school geometry and some 
elementary trigonometry. Let's look at a picture:

  

There is a formula for the measure of each interior angle of a regular 
polygon:

   (n-2) * 180/n

where n is the number of sides. Notice that for a square (n = 4), we 
get 90 degrees, which is what we expected.

When n = 8, we get 135 degrees. So each angle inside the octagon has 
a measure of 135 degrees. The segment that I have drawn that has as 
its endpoints the center of the octagon and one of the vertices of 
the octagon bisects one of the interior angles.  Hence its measure is 
135/2 = 67.5 degrees.

Trigonometry tells us that tan(67.5) = 4/x which implies that

x = 4/tan(67.5) = 1.65685424949

But notice that x is just half the length of a side. So the length of 
a side is actually 2*x = 3.31370849898

I hope this has been clear.  Please write back if you'd like to 
discuss this further.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/