Octagon Side Lengths
Date: 08/22/2001 at 22:26:26 From: Rob Subject: Octagon I need the formula to compute the length of each side of a octagon. I only know the dimension of the octagon from one side to the other, which is 8 feet. (It is 8 ft. from the center of one side to the opposite side.) How can I find the lengths of the 8 sides? I tried to draw it out on graph paper starting with an 8-foot sguare and scaling it with my rule using 22.5-degree angles, but I was not sure about my accuracy.
Date: 08/24/2001 at 08:58:13 From: Doctor Paul Subject: Re: Octagon We can solve your problem using a bit of high school geometry and some elementary trigonometry. Let's look at a picture: There is a formula for the measure of each interior angle of a regular polygon: (n-2) * 180/n where n is the number of sides. Notice that for a square (n = 4), we get 90 degrees, which is what we expected. When n = 8, we get 135 degrees. So each angle inside the octagon has a measure of 135 degrees. The segment that I have drawn that has as its endpoints the center of the octagon and one of the vertices of the octagon bisects one of the interior angles. Hence its measure is 135/2 = 67.5 degrees. Trigonometry tells us that tan(67.5) = 4/x which implies that x = 4/tan(67.5) = 1.65685424949 But notice that x is just half the length of a side. So the length of a side is actually 2*x = 3.31370849898 I hope this has been clear. Please write back if you'd like to discuss this further. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/
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