No-number Geometric Layouts

Many layout problems that seem to call for math can be quickly solved with a compass and a straightedge. Numbers can confuse and complicate the situation. What you need to know is how to make things fit –
not necessarily what numbers describe the lines and angles. Here are a few time-tested methods.





Robert W. Lang; illustrations by Mary Jane Favorite

You may also be interested in “Essential Guide to the Steel Square” by Ken Horner.

9 thoughts on “No-number Geometric Layouts

  1. Great Grey

    Well this was standard draftsman technique before CAD programs. If you can find a good book on drafting it will tell you how to do these and other things by just using a compass, triangles (30-60 and 45) and T-square.

    1. Paul Allison

      Great Gery is right. In High School and Tade School I took four years of Drafting/Mechanical Drawing (1963 – 1967). The closest thing we had to a CAD system was a slide rule.

  2. bsrlee

    The triangle/hexagon method is only approximate – there is always a gap at one corner because pi is not equal to 3, it is roughly 22/7ths. If you step the radii off from one point running around the circle you will find a noticeable gap, stepping off from 2 points reduces the visible error but there is still a gap.

    You need to mark out a 60 degree section first, drawing 2 rays from the centre to the circumference and then then set the compass to the length of the chord formed where the 2 rays intersect the circumference.

    Luckily a 3:4:5 ratio right angled triangle gives you 30 and 60 degreees to go with your more easily found 90.

    1. MStone

      A 3:4:5 Ratio Right Triangle is NOT a 30-60-90 triangle. The ratio of the Hypotenuse to the base of a 30-60-90 is 2 : 1. So if you have a base of 3 you would need a Hypotenuse of 6, not 5. Just look up the Sine of 30, (0.5 or 1 over 2) The Cosecant of 30 (2), The Cosine of 60 (0.5 again) or the Secant of 60 (2 again).

      As a professional stage carpenter, I see people make this mistake ALL the time, so bsrlee shouldn’t feel bad, it is a commonly held belief. But if you’ve ever tried to use a 3-4-5 to do a layout and wondered why you couldn’t get it to work, that was probably it.

    2. Robert W. Lang

      pi has nothing to do with it. If you divide a circle into six equal slices, the angles at the center will be 60 degrees, and the two lines from the center will form an equilateral triangle with the chord as the third side. If the radius length = the chord length then stepping around the circumference works just fine.

  3. john6.0

    Good idea to publish for woodworkers ~~
    Thank You Mr. Lang & Ms. Jane
    I enjoyed reading your return Shannonlove will look for that book.
    Mr. Szczepanski~ Thats a good idea. I wonder if such an illusterated book exist ????

  4. shannonlove

    Working with a compass/divider is almost a lost art today prior to the late 1800s and the development of mass produced, accurately marked metal rulers and the like, everything was measured, marked and even marked using the compass.

    Newton originally created the laws of gravitation using geometry and a compass, only later creating calculus to make the laws numerical. Galileo wrote a great book on using the compass to do numerical calculation. It was basically using the compass to divide lines, making a linear “gear” that could be “meshed” with other “gears” to form a on paper computer.

    The compass is also the reason that the imperial/standard measuring system uses base 12 i.e. feet and inches with divisions of 2,4,8,16,32. That system is easy to use numerically with a compass while the metric system is almost impossible. The metric system didn’t really catch on until the development of mass produced marked tools.

    The great advantage of using a compass is that once you master it, its the only measuring and marking tool you need. That was a great boon back in the day when everything was hauled by muscle power.

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