Constructing angles without a protractor

A student needed to draw a circle with a 2" diameter, then draw the following angles: 100°, 120º, and 140º. She had her compass but didn't have her protractor.
First she drew the circle, then she drew 2 perpendicular diameters. Since a circle encompasses 360º, each quadrant comprising 90º. We drew the 120º angle first using an entire 90º quadrant plus 1/3 of the adjacent quadrant, erasing the unneeded line, which leaves 60º in that second quadrant.
Then we found the circumference of the circle (C=πD, or 3.14x2"=6.28"). Next we found 1/4 of the circumference (6.28"/4=1.57"). We wanted to be able find the arc length in 10º increments, so we divided the arc of one quadrant by 9 (1.57"/9=0.174"). We converted this into 1/16ths of an inch by multiplying by 16 (0.174"x16=2.79 sixteenths of an inch).
Getting back to our angles, we measured the 100º angle next by taking our remaining 60º and adding 40º of arc length to it (4x2.79=11.16 sixteenths). Note: It may be helpful to mark sixteenths of an inch along the arc and reposition the ruler after each. The final 140º angle arises naturally.
Hope this helps if you (or a student) needs to construct angles without a protractor. :)


But this isn't strictly construction in the conventional manner.  It's approximation, based on decimal values. Construction are theoretically exact (the ones based on compass & unmarked straightedge).
The 'weakness' of this presentation is contained in this phrase:  We drew the 120º angle first using an entire 90º quadrant plus 1/3 of the adjacent quadrant, erasing the unneeded line.  Precisely what does it mean to take one third of a quadrant
I agree that it is approximation and not as exact as using a protractor. A quadrant is 1/4 of the circle, or 90 degrees. One-third of 90 is 30 degrees.
A method for the trisection of an angle has yet to be found/presented by anyone!
Actually you can get close by bisecting a 90 degree angle, then bisecting the 45 degree angle, then bisecting the resulting 22.5 degree angle, and adding the resulting 11.75 degree angle to one of the 22.5 degree angles, which brings you 34.25 degrees--not quite 30 degrees, but at least in th ballpark. If up you need to be more exact, you could measure the distance between two points that are on two legs of the angle, equidistant from the vertex, and divide by 1/3.
--Jon the ballpark...


Jon A.

Math, English Tutor

1250+ hours
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