I read once in Dennis Laney's blog another method to octagonize square stock. It turns out that the ratio 7:24 gives an almost exact approximation to find the location of the guide lines.
A little math first - bear with me.
|A square with 1" sides and octagon lines drawn in|
Consider one of the small triangles formed in a corner of the square. As Greg wrote, the Pythagorean Theorem gives us x^2 + x^2 = y^2. We also know that for any side of this square, x + y + x = 1.
With a little algebra, we can combine these two equations and solve for x.
x = 1/[2+sqrt(2)], which is equal to 0.293.
So if you have square stock that is 1" on each side, you can mark lines 0.293" from each edge to find your octagon vertices. For stock larger (or smaller) than 1", you can use the 0.293 as a multiplier to get the proper position. For example, in stock that is 1 1/4" square, the lines for octagonizing will be 1.25 * 0.293, or 0.366" from the edge.
I keep an Excel sheet with decimal equivalents for every 64th of an inch. 0.366" is between 23/64" and 24/64". So I set my marking gauge to that and mark the workpiece.
|Setting gauge to just over 23/64ths|
|24" ruler kitty corner on the stock, pencil pointing to the 7" mark|
On some legs I'm making for a step stool, I've tapered the blanks to the top and to the bottom, with a fat part below center.
|Square tapered legs; 1 1/4" thick at widest part, 7/8" thick at ends|
|Gauge mark locations for ends and middle of the tapered leg|
|Marking the middle at just over 23/64" (but not pulling the gauge along the side)|
|And the gauge marks at the ends at just over 16/64" from the edge|
|Parts marked and ready for planing|
|Four legs tapered and octagonized|