Area of an Octagon

Date: 10/26/2001 at 09:18:06
From: Brad 
Subject: Square footage of a Octagon Shape

Good morning.

I am an insurance agent trying to figure out the square footage of an 
octagon-shaped house. Each wall measures 15 ft. in length. It has been 
years since I have used this type of information. Any suggestion is 
greatly appreciated...

Date: 10/26/2001 at 13:54:08
From: Doctor Paul
Subject: Re: Square footage of a Octagon Shape

I'm assuming you mean a regular octagon.

The measure of each interior angle is 135 degrees. In general, if you 
have a regular n-gon (n sides) then the measure of each interior angle 
is given by:

(n-2) * 180 / n

Note that for n = 4 (a square) we get that the measure of each 
interior angle is 90 degrees (as we should).

If you plug in n = 8, you'll get 135 degrees.

Draw an octagon and label the vertices clockwise one through eight.  
Now connect the following pairs of vertices with straight lines:

1,6
2,5
3,8
4,7

  

Doing this will give you four triangles, four rectangles, and a square 
in the middle. Since the measure of each interior angle is equal to 
135 degrees and the lines we drew create right angles, the smaller 
angle left over must have measure 45 degree. Thus each of the four 
triangles is a 45-45-90 right triangle. Since we know that the 
hypotenuse has length 15, this tells us that the lengths of the sides 
of the triangles are 15/sqrt(2) 

But 15/sqrt(2) is also the length of the shorter side in each of the 
four rectangles.

Thus we have four triangles with base = 15/sqrt(2) and 
height = 15/sqrt(2).

Each triangle gives an area of 1/2 * b * h = 225/4. But we have four 
of them so the area given by the four triangles is 225 square feet.

There are also four rectangles with length = 15 and 
width = 15/sqrt(2).

Each rectangle gives an area of l*w = 225/sqrt(2). But we have four of 
them so the area given by the four rectangles is 900/sqrt(2) square 
feet.

Finally, the square in the middle is 15 by 15 and thus has area 225 
square feet.

Thus the total area is:

   225 + 900/sqrt(2) + 225 =

   450 + 900/sqrt(2) which is approximately equal to:

   1086.39610307 square feet.

For methods for finding the area of an octagon using triangles, see 
the Dr. Math archives:

   Area of a Regular Octagon
   http://mathforum.org/library/drmath/view/54092.html   
   http://mathforum.org/library/drmath/view/54141.html   

If you'd like to talk some more about this, please write back.

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