Area of an OctagonDate: 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/ |
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