Good, we have an octagon in a square - but how good is it? Let's find out! Clearly all the angles are equal, and the lengths of sides 1-2, 3-4, 5-6, and 7-8 are all the same, as are the lengths of sides 2-3, 4-5, 6-7, and 8-1. But are these two groups of sides all the same length? I'll calculate one first, then the other, and see how they compare. You gave a length of 180mm as the length of your square's sides, but for the sake of the argument I will call that value N. Now, when you draw the arcs, the length from the center of the arc to any point on the arc is N*sqrt(2)/2. So the amount outside the arc is N*(1-sqrt(2)/2). Removing the two sections on either side of side 1-2, then, gives us a length of N - N*(2-sqrt(2)), or N*(sqrt(2)-1). Meanwhile, the triangles cut off are right isosceles triangles. The legs are both length N*(1-sqrt(2)/2), meaning the hypotentuse, side 2-3, is N*sqrt(2)*(1-sqrt(2)/2), or N*(sqrt(2)-1). So we have a regular octagon after all! To be honest, I'm actually kind of relieved - I've seen quite a few videos claiming to offer compass-and-straightedge ways for an arbitrary regular polygon to be inscribed in a circle, or to be drawn given a known side - never mind the fact that it's been proved that only certain regular polygons can even be constructed PERIOD using only those two tools!
My son and I just did this with simalar method ... For a homeschooling exam ... Then built a fire safety platform for a kerosene heater to show the applied mathematics at work .
