Stepping off a polygon

[mklotz]

In a recent thread on another forum, someone wanted to make a 14 tooth ratchet and asked for methods for dividing a circle into 14 segments. Since he lacked a rotary table and dividing head, it was suggested that he use dividers to step off chords around the circle. After all, this was good enough for the ancients who made the Antikythera mechanism so it ought to work here.

It was suggested that he guess the initial divider setting and step off around the circle. If he didn't get 14 divisions, adjust the dividers and try again. Continue this iterative process until he obtains exactly 14 divisions.

Now, anyone who knows anything about trigonometry will realize that iteration is about the dumbest way to approach this problem. So how do we calculate the initial spacing of the dividers?

Refer to my canonical diagram of a circular arc below...

A regular (i.e., all sides equal length) polygon of N sides has been inscribed in a circle. The chord BD represents one of these sides. At the center of the circle it subtends an angle of 2x. We'll use the label T for 2x, i.e., T = 2x. Since all the sides are equal we know that T = 360/N.

We now have an isosceles triangle, ABD, where we know an angle, T, and the two equal sides, r, and wish to know the remaining side, BD. Fortunately, trigonometry provides us with a formula, called the "law of cosines", to compute this side. If 'c' is the unknown side opposite the angle 'theta' and 'a' and 'b' are the other two (not necessarily equal) sides, then the law of cosines says...

c^2 = a^2 + b^2 - 2 * a *b * cos(theta)

Applying this to our problem, we have...

(BD)^2 = r^2 + r^2 - 2 * r * r * cos(360/N) = 2 * r^2 * (1 - cos(360/N)

And, solving for BD, the required divider setting, yields...

BD = r * sqrt (2 * (1 - cos(360/N))

Now, as a check, every schoolboy knows that if you step off the circle with the radius used to draw the circle, you'll get a perfect regular hexagon. So, with N = 6 the formula should show us that BD = r. Substituting, we have...

BD = r * sqrt (2 * (1 - cos(360/6)) = r * sqrt (2 * (1 - cos(60)) = r * sqrt (2 * (1 - 0.5)) = r * sqrt (1) = r Q.E.D.

So, armed with this formula, you can compute the required divider setting for stepping off a regular polygon of any number of sides.


--

A little math aside here. Note that if c is the hypotenuse of a right triangle opposite the right angle, theta = 90, the law of cosines reduces to the Pythagorean theorem, c^2 = a^2 + b^2

[mattthie]

Not to be picky, but triangle ABD is an isosceles triangle, not an equilateral triangle, except in the one case where BD=r.

[bricolage]

Stepping off would be faster than the time spent reading this alternative method, let alone trying to understand it or working it out. However, now knowing the math, it might be useful at another time.

[olderdan]

I have never being very good at maths so you could try it my way, using CAD draw a circle of the required segments, radius and measure B and D. After that it comes down to how good you are at setting and using dividers.

[Frank S]

There is nothing wrong with math, my trig started failing me years ago, always good to see the formulas posted. I am also guilty of over using cad to work out my problems instead of hunting down my scientific calculator.

[mklotz]

Not to be picky, but triangle ABD is an isosceles triangle, not an equilateral triangle, except in the one case where BD=r.

Thanks for pointing that out; I've corrected the original post.

[Frank S]

Another way to dived a circle into the number of segments is to use a degree wheel or angle finder compass take the number 360° divide by the number desired note the degrees then set the compass to the desired angle on a center line extending from the center to any point on the circle draw lines from the center along the compass arm set the dividers points to the intersecting points then step off the points and your done. the only math required is simple bonehead division
Marv's way is a lot more eloquent if your not like me who's higher math skills have increasingly failed over the years from disuse

[mklotz]

Another way to dived a circle into the number of segments is to use a degree wheel or angle finder compass take the number 360° divide by the number desired note the degrees then set the compass to the desired angle on a center line extending from the center to any point on the circle draw lines from the center along the compass arm set the dividers points to the intersecting points then step off the points and your done. the only math required is simple bonehead division
Marv's way is a lot more eloquent if your not like me who's higher math skills have increasingly failed over the years from disuse

360/14 = 25.174... deg

A protractor capable of setting that angle accurately is going to cost a lot of money. It's a lot easier to set dividers accurately to a precise linear measurement.

I don't understand what the terror over a bit of math is all about. You simply plug numbers into one simple equation from my original post...

BD = r * sqrt (2 * (1 - cos(360/N))

and you have the required length for an N-sided polygon. Yes, a scientific calculator or equivalent is required but these are all over the web.

[Frank S]

No argument there Marv, but I was 1 of those who still used my slide rule and Smoles to check my scientific calculator well into the 90s about 2000 I got my first computer in 2010 I suffered extreme oxygen deprivation which killed off much of the analytical portion of my brain, which has only recently been partially returning due largely to a lot of your postings

[mklotz]

No argument there Marv, but I was 1 of those who still used my slide rule and Smoles to check my scientific calculator well into the 90s about 2000 I got my first computer in 2010 I suffered extreme oxygen deprivation which killed off much of the analytical portion of my brain, which has only recently been partially returning due largely to a lot of your postings

Checking a scientific calculator with a slide rule? Good one, Frank, I'm still laughing.

Every shop should have a scientific calculator. I regard it as the second most important tool. Today, good ones, e.g. Casio, can be had for less than $20.

I have an HP35S in the shop and one at my desk. A bit more expensive but they're programmable and that's a big plus for me.

If you want to try one yourself, you can download an emulator here...

https://www.educalc.net/2336231.page

and the manual to go with it here...

https://www.educalc.net/2336445.page

Unusual for HP, the HP35S can be switched between RPN and algebraic mode, though why anyone would use the latter is beyond me. :-)