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Side Length of Octagon Inscribed in Square

Date: 9/11/96 at 19:38:16
From: Anonymous
Subject: Side Length of Octagon Inscribed in Square


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!   
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

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