Area of an Octagonal House

Date: 08/24/2003 at 01:24:38
From: Jim
Subject: Octagon house square footage

I plan to build an octagonal home, and would like to know the area of 
the completed home if each wall span is 14 feet in length.

Any information you could offer on this subject would be appreciated 
very much. Thank you!

Date: 08/24/2003 at 04:37:32
From: Doctor Jeremiah
Subject: Re: Octagon house square footage

Hi Jim,

An octagon is eight triangular pieces:

           +------L------+
         +  \           /  +
       +     \         /     +
     +        \       /        +
   +           \     /           +
   |    +       \ a /       +    |
   |         +   \ /   +         |
   |              +              |
   |         +   / \   +         |
   |    +       /   \       +    |
   +           /     \           +
     +        /       \        +
       +     /         \     +
         +  /           \  +
           +-------------+

Each triangle has a side length L and an inside angle a. Here is one 
of the triangles:

           +------L------+  ---
            \           /    |
             \         /     |
              x       x      | H
               \     /       |
                \ a /        |
                 \ /         |
                  +         ---

We know that a = 360/8 and we also know that L = 14 feet. The area of 
this triangle is 1/8 of the area of the whole place.

Any time an angle is involved usually trigonometry is needed to find 
an answer. This problem is no exception. In this case we need the 
trigonometric tangent function. Trigonometry is much easier to use 
with triangles that have a 90-degree angle. If we break our triangle 
in half it will have a 90-degree angle:

           + - - - +---L/2-+
                   |90    /
             \     |     /
                   H    /
               \   |   /
                   |  /
                 \ | <---- angle = a/2
                   |/
                   +

In this case the tangent of the angle (a/2) is equal to the opposite 
side (L/2) divided by the the adjacent-side (H), or written 
mathematically:

  tan(a/2) = (L/2)/H

We know the value of a is 360/8 so:

  tan((360/8)/2) = (L/2)/H
  tan(180/8) = (L/2)/H

And then we solve for H:

  H = L/(2 tan(180/8))
  H = L/(2 tan(22.5))

Now, remember that we said that the area of the big-triangle was the 
length of the base (L) times time-the height (H) divided by 2?  Well, 
we can solve this-by starting with:

  Triangle_Area = (LH)/2

We know the area is 3000/8 and we know something about H so we can 
substitute those in:

  Triangle_Area = (L(L/(2 tan(22.5))))/2
  Triangle_Area = (L L)/(4 tan(22.5))
  Triangle_Area = (L^2)/(4 tan(22.5))

Now, we know that L = 14 so we can say:

  Triangle_Area = (14^2)/(4 tan(22.5))
  Triangle_Area = 118.3 square feet

Now this is the area of each triangle and since there are 8 of them 
the entire place has an area of

  Octagon_Area = 946.37 square feet

Here is a bit more info.  If we can start with a square:

   +  -- D --  +-------L-------+  -- D --  +
             /                   \          
   |       /                       \       |
   D     L                           L     D
   |   /                               \   |
     /                                   \  
   +                                       +
   |                                       |
   |                                       |
   |                                       |
   L                                       L
   |                                       |
   |                                       |
   |                                       |
   +                                       +
     \                                   /  
   |   \                               /   |
   D     L                           L     D
   |       \                       /       |
             \                   /          
   +  -- D --  +-------b-------+  -- D --  +

Notice the triangles in the corners. Because they have a 45-degree 
angle the amount labeled "D" cut off each side of the board is the 
same. But also the long side of the triangle and the uncut part of the 
board must be the same.

The Pythagorean Theorem says that the sum of the squares of the two 
short sides equals the square of the long side. So, using ^2 to mean 
squared, it basically says D^2 + D^2 = L^2, which is the same as 
2D^2 = L^2

Now we know that the length of L is 14 feet but since 2D^2 = L^2 then 
D = 9.9 feet so the area cut off each corner is 49 square feet. If 
this were the whole square the square footage would be 1142.37 square 
feet.

Since the octagon part is 946.37 square feet and the full square would 
be 1142.37 square feet the corners that are cut off account for 17.2% 
of the area of the full square. The octagon accounts for 82.8% of the 
total area.

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

Date: 08/24/2003 at 05:46:41
From: Jim
Subject: Thank you (Octagon house square footage)

Thank you very much for your quick response, and for the excellent 
information you gave me. I appreciate it greatly!