Octagon Formula

Date: 07/30/97 at 21:36:58
From: Charles Ecker
Subject: Octagon Formula 

If you're building an octagon on a 12-foot radius, what is the length 
of each side?   

Thanks :)

Date: 07/30/97 at 22:07:47
From: Doctor Scott
Subject: Re: Octagon Formula 

Hi Charles!

Great question. When you say a "12-foot radius", I assume that you 
mean that the distance from the center of the octagon to a corner of 
the octagon (called a vertex) is 12. Also, from your question, I'm 
assuming that you want to construct a regular octagon with all the 
sides the same length. It might help, too, if you sketch a picture as 
we go along.

Okay, we can drop a segment perpendicular to any side of the octagon 
from the center to create a right triangle (with the radius you 
described). The segment we just dropped is actually called the apothem 
of the octagon. We can now use a little trigonometry to determine the 
"bottom" or third side of the triangle, which is half of the side of 
the octagon.

To use trigonometry, though, we need an angle. We have two options.  
First, we can use the formula for an angle of a regular polygon, 

   angle = [(n - 2) * 180 ] / n, 

where n is the number of sides of the polygon. So, for an octagon, 
each interior angle has measure 135 degrees ( (8-2)*180 / 6 ).  
Now, in the sketch, we can prove that the angle in the triangle is 
HALF of this 135, or 67.5 degrees.  (** A second way is shown below.)

So, we can now use trig:  

  cosine is defined as ADJACENT over HYPOTENUSE.  

We know that the hypotenuse is 12, and we want to find the adjacent 
side, so cos 67.5 = ADJ / 12 or ADJ = 12 * cos 67.5.  Using a 
calculator, I get ADJ = 4.5922.  Remember (you can see it in the 
figure, and prove it using congruent triangles) that this is HALF of 
the original side of the  octagon.  So, each side is 9.1844. (to 4 
decimal places).

Hope this helps! By the way, a second way to find an angle for the 
triangle is to use the idea that we can draw an octagon ON a circle by 
labelling 8 points. That is, we divide the circle into 8 congruent 
arcs, each 45 degrees (360 / 8). If we draw segments from the center 
of the octagon to two adjacent points (and connect the two points), we 
have created an isosceles triangle. The angle at the center, called 
the vertex angle, is 45 degrees.  (From geometry, a central angle has 
the same measure as the arc).  

Draw in the apothem, and each angle at the center will be 22.5 degrees 
(no big suprise - remember that the angles of a triangle add up to 
180, so if one is 90 and one is 62.5, the third must be 22.5).  

Now, sine is defined as OPPOSITE over HYPOTENUSE.  Again, the 
hypotenuse is 12, so, sin 22.5 = OPP / 12, or OPP = 12 * sin 22.5, or 
opposite is 4.5922.  Whew!

-Doctor Scott,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/