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### 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

+  -- 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!
```
Associated Topics:
High School Triangles and Other Polygons
High School Trigonometry

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