Side Length of a Regular Octagon, Without Trigonometry
Date: 09/19/2003 at 17:12:28 From: Gail Subject: Finding the length of the side of an octagon I teach middle school math. A friend asked me how to find the length of one side of a regular octagon that has a (side-to-side) diameter of 16 feet. I was not sure how to go about it, and would appreciate your help.
Date: 09/20/2003 at 02:02:06 From: Doctor Greenie Subject: Re: Finding the length of the side of an octagon Hi, Gail -- Think of the octagon as a square, with the four corners cut off at 45-degree angles. The pieces cut off are 45-45-90 right triangles. Let's use "x" to denote the length of each of the legs of each of these triangles. Then the hypotenuse of each of those triangles is x*sqrt(2). But that hypotenuse is one of the sides of the octagon, so each side of the octagon has length x*sqrt(2) You are calling the diameter of the octagon the distance from side to side; this is the same as the length of the side of the original square. But if we picture the original square with the corner triangles marked but not yet cut off, then each side of that square (i.e., the diameter of the octagon) is made up of three segments in the ratio 1:sqrt(2):1. The side of the octagon is the longer of these three pieces; then the fraction of the whole diameter of the octagon which is the length of one of the sides of the octagon is given by the ratio sqrt(2) ----------- 2 + sqrt(2) So in general, if the diameter of the octagon is x, then the length of the side of the octagon is sqrt(2) x * ----------- 2 + sqrt(2) And in your particular case, with a given diameter of 16, the length of the side of the octagon is sqrt(2) 16 * ----------- 2 + sqrt(2) I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
Date: 09/26/2003 at 20:36:18 From: Gail Subject: Thank you Dr. Greenie, Thank you for your quick response. I appreciate this very much and your explanation was understandable. Thank you again, Gail
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