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### Area of an Octagon

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Date: 10/26/2001 at 09:18:06
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

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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