This is from “Practical Geometry for Builders and Architects” By J.E. Paynter 1921
Here’s a slick way to layout an octagon from a square blank without resorting to math or measurements. Just connect the corners with diagonals and then use a combination square or a small wooden gage block like this one, set to half of one of the diagonals.
George R. Walker
Design Matters 
Thanks. I’m grateful for any shortcut around math (or rulers).
From the illustration, the distance from the blank corner to the nearest point that the octagon intersects the blank side is perpendicular from the blank side to the midpoint of the diagonal running from the same blank corner to center of the blank. This point of intersection on blank side is also equal to the point of intersection of a radius equal to the distance from the blank corner to the center of the blank.
Is this correct?
I’m not sure I follow what you are saying.
George
I was trying to clarify in words what your were showing in the illustration. In other words, get the directions right.
You could also take a compass and put the center point at the center of the original square and the pencil point on the pencil point on the diagram and draw the circle, getting all 8 intersections in one move.
That would be especially handy if you had a tapered blank and needed to know the intersections on each end. Then you could use a straight edge or chalk line to mark the layout along the length.
George
Beautiful. Felt compelled to confirm the geometry and found, for those who may want to figure the blank to end up with a target octagon size, that the final sides will be about 0.4 times the blank width (square root of 2 minus 1, or 0.4142 for those after more precision). So make the blank about 2.5X the desired final octagon width. Thanks, George.
That’s one of the practical things about this simple shop geometry. As you point out with your comment, once you understand you can work a problem backwards or forwards and come up with several ways to apply it. Thanks.
George