On measuring an angle without a protractor
My colleague J stopped in my office today with a curious fact: a technique learned by one of his students to approximate acute angles using nothing but a ruler. According to the student, to approximate the degree measure of an acute angle, simply mark the two sides of the angle at 3 inches from the vertex, measure the distance between these points, and multiply by 20. The result, according to the student, is the degree measure of the angle, within a couple of degrees.
That is, to approximate the measure of an angle, say,
simply measure a distance of 3 inches from the vertex along each ray. The claim is that the 20 times the length of the segment connecting these points is, more or less, the measure of the angle between the rays.
J thought this was a strange result, but he’d tried it on a couple of different angles by hand (measured against a protractor) and was impressed by its accuracy, so he stopped by with one simple question: was this just a happy coincidence, or was there a reasonable explanation for why this trick works so unreasonably well?
I pondered this for a while, and was actually surprised to find a reasonable explanation for this.
In fact, the solution is so pleasing, and raises a few interesting and related issues, that I’m sending it off to be published in the College Journal of Mathematics. So until then, the rest of this post will be unavailable. However, feel free to contact me at travis@komplexify.com if you’re interested in the details.
thanks for this it was just what I was looking for
Comment by peter ellis — 01.3.2007 @
I was looking for measuring angles in a circle without a protracter. I searched that on google and that website came up. This page did NOT have what I was looking for.
Comment by Lindsey — 03.29.2007 @
Sorry Lindsey. As I mention above, I’m trying to get the article published at the CJM. As such, I’m technically not supposed to have the article “published” an some other medium, such as on the web. Nevertheless, if you (or anybody else) is interested, just drop me a line at travis@komplexify.com and I’ll send you a preprint.
Comment by Travis — 03.31.2007 @
this really he;ped me prdict the measurements in my textbook.
to bad i got them all WRONG!
Comment by kayla — 06.3.2007 @
that word was helped not he;ped
Comment by kayla — 06.3.2007 @
Kayla, I’m glad it helped (or he;ped, as the case may be). I’ve submitted my paper to the CJM, so hopefully they’ll like it and I can put the rest of the original post back up.
Comment by Travis — 06.14.2007 @
I remember learning this when I was young & was trying to remember how to do it exactly and have searched every where thanks!
Comment by Jen — 07.10.2007 @
Interesting. I’m trying to measure the angle of my roof for solar heating, so the 3 inch rule is a bit small.
Do you have any thoughts for measuring the angle of such a large object?
Comment by Paul Leese — 03.23.2008 @