Chris Schwarz's Blog

Octagons Made Easy

When I moved from the Deep South to the Midwest, it took me awhile to realize that the word “interesting” is not always a good thing.

For example, Sno-Balls have an interesting aftertaste.

Another example: Woodworkers have a lot of interesting ways of laying out an equilateral octagon.

Why is this important? Hand-tool woodworkers, turners and chairmakers use octagons all the time to make tool handles, to make turnings easier to rough out and to make simple chair legs.

I’ve seen people try to lay them out by eye. This works, but it takes patience and extreme sobriety.

At the other end of the spectrum are people who use the Pythagorean Theorem to calculate each side of the polygon and then lay it out with a ruler.

And I’ve seen people step it off with dividers using trial and error.

The easiest way I’ve found is related to the Pythagorean Theorem, but it uses only one setting of your compass to short-circuit the entire calculation process. It turns a square into a perfect equilateral octagon in seconds.

I use this trick all the time, especially when making tapered octagons, like the legs of a stool I’m building for a future issue of the magazine. The video shows how I do it. If you know a faster way to lay one out on the work, speak up.

— Christopher Schwarz

Hey, I just noticed that the Shop Woodworking store now has “Winchester Desk: Joinery Inside & Out,” a new DVD from Jeff Headley and Steve Hamilton. This is an awesome piece of furniture that these two guys take apart so you can see how everything works. There are lots of surprises.

24 thoughts on “Octagons Made Easy

  1. geppetto425

    Woodsmith magazine had plans a number of years ago for a pencil post bed. They devised a fairly simple jig that slides along your table saw fence to cut the octoganal bed posts, worked great. Still sleeping on that bed.

  2. aschaffter

    As much as I respect Chris, there is a major flaw in his method- it can be difficult to place the point of the compass at the corners- just watch him draw the second arc- it doesn’t meet at the center.

    Easiest, most accurate method:
    1. Draw diagonals
    2. Draw inside circle, tangent to sides
    3. Using a combination square or 45, draw tangents to the circle where it crosses each diagonal.

    Result: perfect octagon; no measuring.

  3. JEBreen

    Another way to lay this out uses a framing square and the “7-17″ rule. Lay the 24″ leg (or any 24″ rule) on a face with one end on one side, the other end flush with the other side. Simply mark the 7″ and 17″ locations and extend your lines parallel to the edges. Of course, this works well for framing or large masts. For small furniture parts, scale down. Use a 12″ rule and mark at 3 1/2″ and 8 1/2″. Or half of that. Knowing the 7:17:24 ratio will get you there.

    I built a small spar gauge using this ratio. Works great. Once its built, you can just grab it and layout your lines, tapered, curves, or straight.

    Jason

  4. DonP

    Chris what Mechanical pencil are you using. I thought you hade talked about them in a former post but could not find it. It looks like it has a long tube that would help in marking out dovetails 
    Thanks
    Don

  5. John Cashman

    I’ve used dividers in this way to lay out octagons for a long time. I’m hoping there will be lots of other uses for dividers spelled out in George Walker’s book when it comes out. If I could only keep one layout tool, it would be a good pair of dividers.

      1. illron

        The spar gauge requires long enough stock for the gauge to register against. If you’re making tool handles out of scrap there’s a good chance you don’t have that extra length. Point for Chris’ method.

        1. wrduffield

          I use a spar gauge that I made. The one in Pete’s link is quite a bit too long to work efficiently for the thickness of the octagon being marked. It is best to make one that is sized appropriately for the octagons you are creating, with the distance between the pins just larger than the maximum thickness of the stock. That way, you can get your layout marks a lot closer to the ends of the blank. Also, the accuracy of the measured ratios is greater when the gauge is closer to perpendicular to the stock.

          I also use mine for marking cabriolet leg blanks after they have been cut out on a bandsaw, and use a spokeshave to take them down from a square to an octagonal profile. In other words, the gauge works for curved octagons as well as straight octagons.

          When you lay out the spar gauge, whether you use a 7:10:7 or 5:7:5 ratio, measure from the center of the pencil, but from the insides of the pins.

        2. JWatriss

          If the gauge is sized appropriately, it’ll register. It doesn’t have to register for the whole piece, just enough to get a line that your straight edge can finish up. Point for spar gauge.

    1. PhilS

      How does the spar gauge address tapered stock? I thought the dowels and pencils are used to align to the stock thickness and then draw the lines respectively.

      Both the dowels and pencils look to be fixed. If that’s the case, how do I draw a line that tapers as I run the gauge the length of the square stock? What if the stock isn’t square? That is, what if I’m using some irregular shaped remnants to make my tapered legs?

      Also the gauge consistently cuts its predetermined ratio so it you want something else aren’t you back to the dividers or making another gauge?

      1. William Duffield

        The spar gauge works just fine on tapered stock. You just apply twist to the gauge until both pins contact the stock, and start drawing. If you grab it near the middle with one hand, then you have to continue to twist; If you grab it at both ends with two hands, then you apply pressure towards the stock with both hands.

        If the stock is not square, you still get an octagon, but it is not a regular octagon. The eight new faces are proportional to the original four faces. If you continue to fair it, you end up with a spar with an elliptical instead of a circular cross section.

        If you start with irregular stock, and don’t want to take the essential step of first making it regular, then ANY method you use will result in irregular (but only maybe) tapered legs.

        If you want something other than a regular octagon, you have to go back further than the layout with the dividers. Chris’s method is also designed for regular octagons only, starting with squares (regular quadrilaterals). If you want a hexagon, or a heptagon or a nonagon, for example, you need to go back to your basic geometry textbook.

        1. PhilS

          When you say “apply twist” is the assumption that the stock is square and the twist is applied to move the pencils progressively inward as the gauge is drawn down the piece? Or, are you saying that one applies twist to keep the dowels consistently along the sides of the stock? If its the latter, doesn’t that assume that the four sides of the stock have already been tapered producing a tapered rectangle?

          Using dividers, its possible to use irregular stock and produce a regular shape I think. Scribe the octagon (or shape of your choosing) on the top and bottom. Then, use a straight edge to connect the corresponding points on the piece. Remove the waste using a hand plane, band saw, etc. You probably need to square the ends (make them parallel) but you can get a regular shape without having to square up or taper sides of the piece first.

          Going back to Geometry shouldn’t be an issue although I suspect there’s a similar algorithm for other regular or symmetric shapes. I would think its all a matter of bisecting the lines correctly. While I haven’t verified this, I would expect there to be similar solutions for other symmetric shapes.

          1. William Duffield

            If you twist the gauge to keep the dowels rubbing against the sides of the stock, and if the stock is tapered, then the pencils move progressively inwards, or outwards, depending on which direction you move the gauge.

            The only thing that is important is to start with a blank with two pairs of mutually perpendicular faces. Whether it is straight, or tapered, or even curved (like a cabriolet leg), you end up with a pair of lines on each face. If you then remove the corners up to the lines, the four new facets formed end up the same width as what is left of the facets you started with.

            You can’t make a silk purse out of a sow’s ear. So, no matter how you try to lay it out, it is imperative that you start with four fair faces.

            If you are really interested, and don’t understand it from our explanations and the diagrams in the link that Pete provided, just build one and try it out.

            1. PhilS

              I understand completely.

              Using a spar gauge, I will taper the already fair piece to accommodate the offset of the gauge, I will use the gauge, and then I will taper again working down to the lines just drawn by the spar gauge.

              Using a compass and a straight edge, I need two adjacent fair sides to align the rectangle at top and bottom, a compass, and a straight edge to align the sides scribed with the compass.

              I draw the two rectangles at top and bottom using the two adjacent fair sides (maybe an off-cut?) and then scribe the shape as the video shows. I then mark the lines using a straight edge. I cut to the lines marked form the rectangle and plane the edges to produce the octagon.

              So, I do more marking with the compass and straight edge but less pre-work getting the piece ready. I can also cut to the ratio of my choosing instead of the one specified by the spar gauge.

              So while it is true that one cannot make a silk purse from a sow’s ear, one can make a silk purse from silk woven by silk worms if one knows the steps to follow.

              Anyway, enough of this. Thank you for your time and explanation and good luck with your woodworking.

  6. John Hutchinson

    Very nice, Chris. I’ve been using lick-and-stick patterns out of my computer for so long that I’ve forgotten some of the basic principles of geometry. Your videos are looking better all the time. Some time soon, you’ll have to give us a quick rundown on your camera, lighting and sound.

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