Perfect Octagons from Squares with the “Wallace Factor”

 

In geometry, a “Regular Polygon” is defined as a convex polygon (polygon is a closed figure formed by a finite number of coplanar segments) with all sides congruent and all angles congruent.  So, a star is not ‘regular’ because it does have concave and convex alternating at every vertex.  Generally, we deal with squares and equilateral triangles.

My last project was to make an Octagon, and I did want all the sides to be equal.  As I played with the measure of the angles and the length of the sides, I realized that it is much better to start with a square and then cut the corners off such that I am left with the octagon shape as I wanted it.  I thought that if I start with a square, then I will have the project under control.  I went to the table saw and cut a piece of plywood to be a square.  Next, I reset my miter to 45 degrees instead of 90.  Now the only thing was to measure from the fence to the saw blade, so that after I cut two of the corners off, I will have three sides of my octagon.  The problem was to determine just what the measure from the fence to the saw blade had to be.  So, I worked for a couple of days and came up with a constant number that calculates the measure I was needing (“e”) as it relates to the length of the side of the square that I was cutting.  I like to call this constant number the “Wallace Factor”.  Here is the work sheet that I developed with Microsoft Excel.

Since I used Excel, I was able to put the calculations within a cell, and that saves you from having to go through all the work for yourself.  Note that in the rectangle that contains ENTER “x”, all you have to do is enter the length of the side of the square and the measure of “e” will be calculated immediately for you.  At the bottom of the page, it automatically calculates ‘a’, ‘b’, and ‘e’, then it does double check the numbers by adding “a + b + a” which is the measure of the side of the square. 

NOTE OF CAUSION:  The measure from the fence to the blade must be made to the correct side of the blade which is the left side of the blade, since the blade width is also part of what you are removing.

The Wallace Factor equals “.207119737”.  If you multiply this .207119737 times the length of a side of the square, you will have the measure, perpendicular to the cut, to remove from the square’s corner. 

If you would like to have the “file” that I created (the photo worksheet shown above), then just email me (stirlingbay@gmail.com) and I will attach the worksheet in the reply back to you for you to have for future work.