I was flipping through a neighbor's library and opened a book called
"101 DIY Projects" or something like it, and there was a good way to
lay out an octagon (it was for a picnic table). Here goes:
1. Lay out a square, and measure the diagonals. E.g., our square has
sides of about 4 1/2 feet and diagonals of 6 feet exactly.
2. Measure and mark 3 feet (half the diagonal) from both sides of
each corner, for a total of 8 measurements and marks.
3. Connect the dots, cut off the corners, and wahlah (sic)! You've
got a perfect octagon!
I usually overthink things like this, so it struck me as particularly
simple and helpful.