An Old Trick (for turning squares into octagons)

Anyone who has laid out an octagon on a long square blank using a set of dividers, knows that it would be a lot easier if humans had evolved with three hands.  And there are times, as in “posts in place”, shaving fifty leg blanks or an octagonal section placed between two square sections, where laying out on an end is next to impossible.  And, then again, you might never have the need for this old carpenters trick.  But, here it is, anyway.

On larger square stock, lay a 24″ framing square or rule diagonally from one edge to the other.  “Tick” mark at 7″ and 17″.  Hand hold a pencil and mark the lines.  (Most people are amazed at how accurately a “hand-held” line can be drawn.)

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For smaller stock, you can use a 6″ or 12″ scale.  Lay the diagonal line from 0″ to 6″. Then “tick” at 1 3/4″ and 4 1/4″.  Follow the path above.

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You’ll be amazed at the accuracy of this method, not to mention the ease.

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9 Comments on “An Old Trick (for turning squares into octagons)”


  1. Nifty trick. How did I not know this? Thanks for sharing.


  2. Reblogged this on Paleotool's Weblog and commented:
    An excellent “trick of the trade.” We could all use three hands when laying out.


  3. I saw a post somewhere on the internet where a writer expressed his bemusement at how the old time textbooks on carpentry and woodworking always started with a long section on geometry.

    This is the kind of time-saving trick that people used to do back before calculators. I consider it a wonderful application of applied trigonometry.

    In case anyone is interested in the math – 1 3/4 is 29% of 6 and 4 1/4 is 71%. So your sides are now divided into three sections, one 29% of the width, one 42% and the third 29% (1 – .71).

    If for example you are using 1” square stock – each corner of the stock now has a .29” section on each end with a .42” section in the middle. The two .29” sections on adjoining sides make a right triangle, so it is easy to determine the length of the hypotenuse, which will be the length of the new side you will cut. .29 squared is .0841, double that and take the square root, and you get .41. So the octagon has alternating sides of .41 and .42 inches – close enough for government work.

    • D.B. Laney Says:

      A fellow “geometry-phile”. You’re absolutely right. Trade books always had sections on practical geometry. We’ve become so dependent by calculus based formulas (derived by others, more often than not), that we’ve allowed the art of geometry to slip away from us. It is such a gift as it allows the maker, the builder, the artist to create, without limits. If you’ve been reading this blog for a while, you know that geometry is a particular interest of mine. If you’re new to the blog, you’ll probably be surprised at how much geometry enters my thoughts and my posts. Pay special attention to anything discussing l’art du trait. The French have got it right.

      Thanks for your comment. There’s just so damned many ways to “skin that old cat”.


    • That’s interesting, but how is it relevant? The simple solution to the octagon is in dividing it into 24 parts, not 100. Trigonometry works beautifully on a digital scale, but much of geometry (and carpentry and architecture) depend on whole-number proportions.
      The ability to divide into thirds, for instance, is integral in inch-foot measurement, not in metric.

      • Sylvain Says:

        You can take whatever length you like, report it 24 times on a wood stick and use the trick. So you can use a length of 24 cm if you have a metric rule.
        Taking a length of 65¿ and making a mark at 19¿ and at 46¿ would give even better result ( ¿ being whatever you like). Would it justify a system where you can divide by 13 and by 5?
        7/24 is as awkward as 29/100 ( in fact more for me).
        Sylvain


      • Metric may have its virtues (dunno, I simply don’t use it), but proportion isn’t one of them.
        7/24 is simply a keystroke, no more or less difficult than any other proportion. Still, does nothing to enhance our understanding of traditional layout and design. (society Villard de Honnecourt has several academic papers on that subject, if it interests you)
        One advantage of the 7/24 layout is that the inaccuracy is outside our theoretical octagon by a shaving or two. Interesting that 65 is a cognate, because with 24, the inner triangles of the octagon are 5-12-13…
        “Not everything that matters can be counted, not every thing that can be counted, matters.”


  4. Well said Michael! The numbers are independent of the geometry.


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