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We all know how to use a protractor; it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch.

For instance, was the understanding of $\pi$ and a compass first required before the first protractor, and if so, how can I draw a full protractor on paper with just a compass, a ruler and some understanding of $\pi$?

I guess my point is, if we can draw a semi-circle on paper, then how can we fill up all degrees without the help of a protractor?

I consider the fact that I can build the tool that I would use to solve the problem in a different way as a great engineering learning experience. I am especially interested in answers that take this aspect into account.

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    $\begingroup$ This a great question! However, it is not a question about math education, it is a question about mathematics, so I think it would be better suited to math.stackexchange.com $\endgroup$ Sep 8 at 17:30
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    $\begingroup$ OP, you could improve this question by mentioning how this will help you to teach. $\endgroup$
    – Sue VanHattum
    Sep 8 at 20:23
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    $\begingroup$ @YGranja are you sure it was rejected? It looks open to me: math.stackexchange.com/questions/4527082/… $\endgroup$ Sep 8 at 21:15
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    $\begingroup$ Why did you ask essentially the same question in Math.SE also? I'm fairly sure that is, if not against the rules, then at least not recommended (and also a bit rude). $\endgroup$ Sep 9 at 10:28
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    $\begingroup$ @Jyrki Lahtonen: It appears not to be straight crossposting, but instead first posting there and being recommended to post here instead. $\endgroup$ Sep 9 at 10:50

5 Answers 5

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As Will Orrick says in the comments under user20315's answer, it is possible, with straightedge and compass, to construct a regular 120-gon, and therefore it is possible to mark off every 3 degrees on a circle.

Can we get any farther? It depends on how much precision you require, and how much error you are willing to tolerate. In principle, it is not possible to trisect a $3^\circ$ angle using only a compass and straightedge. However, the following incorrect trisection method produces angles that are very, very close to correct:

  • Let $O$ be the center of a circle, and let $A, B$ be points on the circle such that arc $AB$ measures 3 degrees.
  • Join $A$ to $B$ to create segment $\overline{AB}$.
  • Trisect $\overline{AB}$ using a compass and straightedge, finding points $C, D \in \overline{AB}$ with $AC = CD = DB$.
  • Draw rays $\overrightarrow{AC}$ and $\overrightarrow{AD}$.

The resulting angles $\angle AOC, \angle COD, \angle DOB$ are not exactly 1 degree each, but the difference between the actual measures and the desired measures are less than 1 part in 10,000. Given the imprecision involved in using mechanical construction tools (how thick is the tip of your pencil? how smoothly can you draw an arc with a compass? how 'straight' is your straightedge?), and the inherent limits involved in reading or using a protractor (can you even measure a degree to less than 0.1 degree precision with a protractor anyway?), this would seem to be good enough for almost all conceivable purposes.

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    $\begingroup$ If you are allowed to draw on the straightedge, you can trisect an angle. $\endgroup$
    – Mark
    Sep 9 at 0:53
  • $\begingroup$ Or if you have a set square (Bieberbach's trisection). $\endgroup$
    – TLW
    Sep 10 at 18:48
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Step 1: Find a machinable material that is reasonably incompressible.
Step 2. Find a string material that is reasonably non-stretchy.
Step 3: Make a cylinder out of the machinable material (perhaps using a lathe).
Step 4: Make a fine groove in the cylinder, of even depth and non-wavering axial elevation. (The lathe can do this, too.)
Step 5: Wrap the string around the cylinder (in the groove).
Step 6: Cut off exactly one lap of the string.
Step 7: Measure off however many even increments you want along the string. (You can do this by making a set of evenly-spaced parallel lines, and placing the taut string between points on the far parallel lines. The number of parallel lines needs to be one more than the number of increments.)
Step 8: Put the marked-off string back in the groove.
Step 9: Optionally, secure the string in place.

Note that steps 1-5 are equivalent to using a compass to draw a circle on a paper. Step 6 is a way to straighten out the circle back to a line segment. Steps 5 and 8 are a way to wrap a line segment into a circle.

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In mweiss' method, the outer angles ∠AOC and ∠DOB are shy of 1° by slightly more than 0.0001°, and the inner angle ∠COD exceeds 1° by slightly more than 0.0002°. On a (huge!) one meter diameter protractor, that is a worst case error of less than 4 microns, or one-seventh of a thousandth of an inch.

Formulas used to calculate the "shy-ness" of the outer angles:
α = angle being trisected
inner angle = 2 * arctan(tan(α/2)/3)
shy-ness = arctan(tan(α/2)/3) - α / 6

If that accuracy is not good enough for you, you are welcome to repeat the process:

  • Mirror image ∠DOB about the line OD,
  • to make an angle ∠AOE that exceeds 1° by slightly more than 0.0002°.
  • Angle ∠COE is slightly more than 0.0003°.
  • Use mweiss' method to approximately trisect angle ∠COE.
  • Let "F" be the new point that is one-third of the way from C to E.
  • Angle ∠AOF is our second approximation of a degree.

Using the previous formula, and impossibly accurate compasses and straight-edges, angle ∠AOF is shy of 1° by slightly more than 10-16 degrees. On a protractor the size of the Earth, 90° would be 10,000 kilometers. Even at that scale, angle ∠AOF would only be shy of 1° by slightly more than 10-11 meters.

In our quantum-mechanical world, you cannot draw anything that accurately. "Solid" matter made of nuclei and electrons gets really fuzzy at a scale of 10-10 meters. If you try measuring any smaller than this scale, you cannot know whether an electron will be where you are prodding or not. So this second approximation of a degree is limited by Earthly physics: you cannot make a compass or a straight edge perfect enough to achieve it, and even if you could, you could not measure how perfectly you drew it.

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In terms of doing it on paper, it is impossible to trisect (or quintisect) angles with a conventional compass and straight edge. And 180 = 2 * 2 * 3 * 3 * 5. Of course you could be an anti-Babylonian rebel and create your own angle measurements on a power of 2.

However, you can use a quadratitrix compass (essentially a moving plane device) for trisection (and quintisection) of angles, to satisfy the conventional degree measurement. See https://en.wikipedia.org/wiki/Quadratrix_of_Hippias for a very short discussion.

P.s. Somewhat tangential, but of interest...see this video on origins of metrology for practical machining. https://www.youtube.com/watch?v=gNRnrn5DE58

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    $\begingroup$ @WillOrrick While it is certainly true that one cannot trisect a $3^\circ$ angle exactly using a compass and straightedge, the most straightforward incorrect trisection algorithm produces a result that is correct to about 1 part in 10,000, which is certainly good enough for most practical purposes. Given the inevitable imprecision involved in using paper and pencil, it's unlikely the other steps in the construction produce results that are any more reliable. $\endgroup$
    – mweiss
    Sep 8 at 16:08
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    $\begingroup$ You can trisect an angle, just not restricted to that one specific set of tools (compass and straightedge). (For instance: it's fairly straightforward to do so if you have a set square, via Bieberbach's trisection.) $\endgroup$
    – TLW
    Sep 10 at 18:47
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    $\begingroup$ Compass and ruler I believe was the phrase. You can trisect with a marked ruler. Also, many - albeit not all - rulers have a right angle at the end. $\endgroup$
    – TLW
    Sep 10 at 19:01
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Use a sine bar and some gauge blocks to machine a line in a the protractor. Repeat many times.

sine bar

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