Side Length of Octagon Inscribed in SquareDate: 9/11/96 at 19:38:16 From: Anonymous Subject: Side Length of Octagon Inscribed in Square Hello, I have a piece of plywood (48"x48") and I would like to cut an octagon out of the middle. How do I get all eight sides equal? What calculations are needed? Date: 9/12/96 at 0:54:42 From: Doctor Pete Subject: Re: Side Length of Octagon Inscribed in Square [Diagram supplied by Dr. Chuck] ----------------------------- | / \ | | / \ | | / \ | |/ \| <--- This triangle is a 45 degree | | right triangle, as are the | OCTAGON | other triangles. | | | | |\ /| | \ / | | \ / | | \ / | ----------------------------- If you cut the octagon out of the square in the "obvious" way, then you will observe that you need to cut the 4 corners of the square at a 45-degree angle such that the length of each remaining side is equal to the length of the 4 cuts made. Say the length of your square is 48 inches. Then from one corner, cut a triangle so that each leg is x inches; that is, you start cutting x inches away from the corner of the square. Then the length of the cut you made is the hypotenuse of this triangle, which is x*Sqrt[2] (x times the square root of 2). This is because of the Pythagorean Theorem, which states that in a right triangle, a^2 + b^2 = c^2, or the square of the hypotenuse is equal to the sum of the squares of each leg. Now, if you cut the remaining 3 corners, you will find that the length of the square's side has decreased by 2x inches; since this is now a side of the octagon, you therefore have x*Sqrt[2] = 48-2x. Solving for x, we see that x = 48/(2+Sqrt[2]), or 14.058875 inches. Thus, you should begin your cut about 14 inches away from each corner, going at a 45-degree angle to the side. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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