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.
Update. 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.
Travis,
I couldn’t think of where I might have “stored” my protractor and other tools from my high school mechanical drawing class, 1962[ big smily face goes here]. Have had a an idea of recent but was unable to get a sense of what I would be requiring in a technical sense and needed the info to talk cogently to suppliers. BINGO, you provided the simple solution. Ah the perspective of the genius mind.
It really is important because if I don’t get the angle close there is a chance my time torquing machine might propel me 250 billion miles into space rather then the mere 238,000 miles I hope to gain. The difference in error multiplies exponentially the distance traveled. Rather touchy stuff. I am also worried about getting caught in a time collapse, you know, solar wind and all. It is interesting how the whole contraption works. To describe it in the thumb nail perspective I call it the “pulling of the leg”. Forgive me for this big tease. But I am a habitual teaser. Hope it brought you a smile, and who knows, maybe you will transduce it into a billion dollar trilogy. Thanks again for giving me the Right Angle.
Bill @ bwdantle@yahoo.
Like the ‘teaser’ above I have mislaid all protractors, and whilst I had roughly estimated the pitch of my roof by drawing the angle, it was nice to have it confirmed by your very useful method. Thanks.
@___@ Does this really work?
Thanks for this fabulous tip – my son left his protractor at school and had homework due the next day. We made a pie graph from scratch using this info. A circle and then another circle 3 inches larger. Divided the required angle by 20 and then measured on the wider circle. It worked perfectly. Thanks again!
this proof is trivial,
assume that x is the distance of that measured side in inches and if we express the angle A (let) in radians then putting the distances of other two sides at 3 inches each we can easily calculate using trigonometry to show that according to your calculations ; i.e. multiplying x by 20 and transforming the angle in radians we get [sin (A/2)/(A/2)] = 3/pi= 3/3.1415928…. which is pretty close to one, but we know sin (A/2)/(A/2) itself is nearly 1 when A/2 is an acute angle.
Largest error occurs when A=90 deg; then sin (A/2)/(A/2)= 0.9003 and 3/pi=0.954; so largest error possible is about 6% i.e. about 5.5 degrees for a right angle.
Fine!!!!!
wow
Except that’s not a proof. You’ve demonstrated that it is a reasonable approximation for a few values.
You would have to prove that A=20x2rsin(A/2) for r=3 ie: A=120sin(A/2)
where the length of a chord is known to be 2r sin(A/2) from other methods.
It’s not true if there exists a single value A, for which the formula does not hold.
Say A=90
then 120sin(A/2) = 120 sin(45). We know that sin 45 = pi/4
so 120(sin(45)) is 120pi/4 = 30pi
if 90=30pi, then pi=90/30 = 3 which we know is false.
Therefore it is not true.
However, this does not mean that this cannot be a useful result, it’s just not true.
oops sin 45 = sqrt(2)/2 but the two sides still do not equal
thxz for the tip
thxz for the help
Thanks for introducing to new method of measuring angles in absence of protractor. I needed to look out for a method as I was helping out my daughter in her homework and she had misplaced her protractor.It works fine.
This worked just great. Was making measurements for baseboards after installing hard wood floors and had no protractor to use for mitre saw cuts that were off of 90 degrees or 45 degrees. Worked like a charm. Thanks so much for this wonderful mathematical tip.
Thanks so much for this wonderful mathematical tip. Came is very handy when I needed to figure out odd angles for baseboards after installing hardwood floors.
your welcome:p
i do not under stand this help
im a math teacher/super weiredo nerd and this does not work and has been proved at the senior school/ collage and sixthform school i work out. and fyi, inches are not used anymore unless you are an over 65
Thats cute Titch, but perhaps a math teacher such as yourself should work harder on your spelling and grammar – a man cannot live on numbers alone, and you risk looking like a retard in front of your “school teacher” peers.
Grammar troll is trolling.
explain more. i got some of it but it was to confusing
Thanks
OK, I get the angle part if you have an idea of where the legs are but what if I know the angle (2 degrees) and one side (90 degrees), how to I create the second side without a protractor?
Is there an App for this?
As an FYI, there is an App, for iPhone’s, but it does not make use of lying flat on a drawing board. The ones I saw are more of an inclinometer.