How to build a protractor without a protractor?

Question

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 the degrees without the help of a protractor?

Answer 1

I think there are two questions here: the practical question of what is actually done at a protractor factory, and the theoretical question of can you decompose a circle into $360$ equal pieces given only a straight-edge and compass.

I'll focus on the latter since the former is not really about mathematics. We know that $360 = 2^3\cdot3^2\cdot5$. Now, $72=2^3\cdot3^2$ degrees is a constructible angle, because a pentagon is constructive. Bisection is always possible, so that leaves angles that need to be trisected twice. This isn't possible with a straight-edge and compass (in general), BUT arbitrary trisection is possible with a ruler and compass (i.e. putting distances on your straight-edge is enough to over-come this hurdle). Wikipedia says this was already known to Archimedes.

Answer 2

For practical protractor production, take an image of a master protractor and print it on paper or plastic.

But I assume that what you actually want to know is: How do you construct a 1° angle? So that you can mark that “master” protractor from scratch.

Start by constructing two shapes:

• An equilateral triangle. As you know, it has 60° interior angles. Bisect it to make a 30° angle.
• A regular pentagon. It has 108° interior angles. Bisect it twice to make a 27° angle.

Use these angles to construct a $30° - 27° = 3°$ angle.

Now, we just need to trisect 3° to make 1°. Unfortunately, it turns out that you can't do that with compass and straightedge. But you do have a few options here:

• Neusis construction, origami, or any known technique that can exactly trisect an arbitrary angle.
• Approximation.
  • Construct a 63° angle (you already have 60° and 3° available from the previous steps), and bisect it 6 times to make a $\frac{63}{64}°$ angle.
  • Or use the identity $\frac{1}{3} = \sum_{k=1}^\infty \frac{1}{4^k}$. Bisect your 3° angle twice to make a $\frac{3}{4}°$ angle, then repeatedly bisect it, adding every second bisected angle to your approximation until you're as close as you need to 1°.
• Just eyeball it. What do you need to measure angles for anyway? The trajectory of a manned rocket to Mars? Positioning a scalpel for robot-assisted brain surgery? Or some personal craft projects? Assuming it's the latter, being off by a small fraction of a degree probably won't hurt.

Answer 3

Since this question was cross-posted at MSE, I am reproducing (and slightly adapting) the answer that I posted there:

It is possible, with just 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.