How do you make an Octagon from a 5 inch square board?

This is just a partial because I am looking for the same answer. Start with a 5 by 5 square. Cut each corner with a line that we call X. The space left on each square side between where the lines we drew intersect the square side is the same length, therefore we call that segment "X" also. Now the segments left over are the legs of each of the four triangles and they are equal in length, so that we can call them each "A". We have two equations from this. Pythagorean's theorem A squared plus A squared equals X squared, which is two A squared equals X squared. And then 2A plus X equals 5. If you solve for either then plug in the answer the problem is solved but my equation abilities are 20 plus years gone so that's where I'm at.

My equation abilities are long gone also. How I accomplish this is to create a square that is the same size as what the largest dimension of the octagon will be, point to point. Then I intersect the square corner to corner, then from the center of each side. This forms an "x" and a "+". Measure the distance from the center to the outside of the "+". Then run that measurement from the center out each of the "x" lines. Connect the ends of the "x" and "+" lines and you have an octagon with equal sides. This works as well laying out a gazebo as it does on paper.