How to build a protractor without a protractor? 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?

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:

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*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.

