## More trigonometry, this time for shop built knobs

Today I present another example of applied trigonometry in the woodshop. I was curious how I could make knobs for shop jigs and avoid purchasing those black plastic jobbies. Having the ability to make whatever size and type I needed versus having to special order knobs appealed to me. I figured it would involve a lathe, which I don’t own, but I was hoping for another solution and I quickly found it using Google. Star knobs!

Your basic star knob is made by drawing a circle on flat stock, then dividing the circle into equidistant lengths along the circumference. A Forstner bit of appropriate diameter can then be used to “take a bite” out of the circle at the beginning of each of these lengths. The knob can then be cut free with a band saw, fret saw or jig saw. The rest of the knob shaping must be done by hand. I used a belt sander and a drill press drum sanding attachment to refine the shape. There are some wonderful and insightful videos on this topic on You Tube.

I have only made these knobs on a single occasion as part of the build for my bench top router table. In this example I used my dividers and geometry to divide the circle into equal parts. You can get close just eyeballing it and even closer using your dividers with trial and error. It occurred to me that I should be able to calculate the length of the segments that I need based only on the diameter of the circle and the number of stars that I want on my knob. This turned out to be true and I wanted to share the reasoning.

One way to solve the problem is to simply divide the total circle circumference by the number of desired segments. Let’s assume my diameter to be 2″ and I have selected 5 knobs. We know that the length of the circumference is equal to the diameter multiplied by the irrational constant π which can be approximated as 3.14.

c=πd=2″*3.14 =6.28″

We can now divide the total length 6.28″ by the number of segments 5 to arrive at the length of each arc 1.256″.

Now my problem is that I want to use my dividers to do this and they don’t help me measure arcs. So what I’d really like to know is the cord length between these arcs. Before we go further let’s look at some of the definitions.

In the last paragraph we calculated the arc AB length to be 1.256″. What I really need to determine is the length AB that I can measure with my dividers. So my known values are the lengths of both sides AO and BO which is the radius of the circle. If the diameter is 2″ than the radius is half that or 1″. I can infer the inscribed angle AOB because I know that any circle has 360° which I will divide into five equal parts. Thus angle AOB is 360°/5 stars =72°/star in this example.

I can’t use the tangent function because this is unfortunately not a right triangle (meaning none of the included angles of the triangle are 90°). I can however, use a formula known as The Law of Cosines which states:

*In any triangle, given two sides and the included angle, the third side is given by the Law of Cosines formula: c*^{2} = a^{2} + b^{2} – 2ab cos(C)

In my example a is AO, b is BO and C is the inscribed angle. AB is the chord length I am after. Now it’s plug, play and calculate.

AB^{2}= 1^{2} + 1^{2} – 2*1*1 cos(72°) = 2 – (2*0.309) = 1.382 “^{2} (units are square inches)

AB=√1.382″^{2} = 1.176″

So there it is. Now I can draw my 2″ diameter circle and using my calipers, I can approximate 1.176″ and begin marking my chords around the circle. If I want, I can then do the same procedure marking in the other direction to see if the points exactly coincide. They may not but they should be quite close.

You shouldn’t be surprised that I made a simple spreadsheet widget to calculate these values. Who knows, I might decide to make knobs in other diameters and shapes?

Click on the picture to play with or download the calculator.

Armed with this information you really don’t need to mess with my widget, there are many circle/arc/chord calculators online and they are easy to find now that you know they exist and what they are called. The terminology can be confusing when using these online resources so be sure to “take care of the units (of measure) and the units will take care of you!”.

*You have been reading an excerpt from the shop journal of the Turtlecovebrewer.*