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.

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.

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25 Responses to On measuring an angle without a protractor

  1. Bill says:

    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.

  2. 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.

  3. Len-Chan says:

    @___@ Does this really work?

  4. Annie says:

    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!

  5. chiranjib says:

    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!!!!!

  6. jade says:

    thxz for the tip 😉

  7. jade says:

    thxz for the help

  8. ajay says:

    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.

  9. 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉 😉

  10. Linda Hyatt says:

    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. 🙂

  11. Linda Hyatt says:

    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.

  12. abbie says:

    your welcome:p

  13. linda powell says:

    i do not under stand this help

  14. titch says:

    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

    • God Botherer says:

      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.

  15. tristan says:

    what if the line is shorter than 3 inches

  16. ken says:

    Thanks for the tip works really well. I experimented with it, and it seems that the explanation for this method may very well be that, by measuring at 3 inches, provides two equal right triangle-sides that can then be calculated as a 60 degree triangle. If you think about the fact that 3 x 20 = 60, it’s the perfect formula for a perfect triangle and by making an angle at 3″ gives us two equal sides in length, same as if we were to calculate A+B+C = Degrees. Being the case that it is the “Angle” that is to be found here, one could simply find the angle using the 3″ equals, and then simply re-scale to bigger or smaller porportions. Bravo sir, well done giving us this neat tip.

  17. Pingback: /954

  18. George says:

    Guys, I have made my own formula for this, but it only works on right-angle triangles.
    The formulas is;
    n/2 = x
    45/x = y
    yb = z

    z = the angle.
    Say you have a length of 50 and a width of 20. Make your right angle, and then connect the ends of the arms to create c. (c) is the hypotenuse. You do;
    a^2 + b^2 = c^2. a = 50. b = 20. 50^2 + 20^2 = c^2. This equals 2500 + 400 = 2900. If 2900 is c^2, you have to square root it to equal c (sqrt(n^2) = n). This equals 53.85164807134507104915403. Apply that number to your unit of measurement and that is how long c equals. This is called the “Pythagoras’ theorem”. But now, onto my own formula. Do this;
    c – a = t
    You then get how much c has moved through 20 in width. (t) is just a way of representing the answer. Knowing that, you divide t by b (which equals 20).
    t/b = (how much c has expanded its length through 1 unit of width, which is 0.19258240356725201562535524577016 units in measurement).
    Now that you have this, you do the first formulas I showed you. Say you want to find the angle between a, c. You do;
    a/2 = x
    a = 50. 50/2 = 25. 25 = x.
    Then you do this;
    45/x. This equals 45/25. Why do this? Because if the width was to be half the length, and you were to connect both arms of your right angle, you would have a 45 degree angle instead. If you were to have the same length as width, we do not know exactly what angle you will have (accurately). All we know is that c would equal sqrt(2), which is irrational.
    Now doing 45/25, that equals 1.8. You have your angle that is created through 1 unit of width, meaning that every time c has to expand; 0.19258240356725201562535524577016 units in measurement,
    you get 1.8 degrees. So now what do you do? Obviously, you multiply 1.8 by b (which is 20).
    1.8 x 20 = 36 (this represents yb = z). The angle has 36 degrees. Don’t believe me? Look at this;
    0 5 10 15 20 25. This is width.
    0 9 18 27 36 45. This is the degree of the angle.
    I just simplified the pattern by fives, because 25 is a perfect squared number (square root 25 and you get 5, because 5 x 5 = 25), but you can count from 1 – 25 in width, multiplying each number by 1.8 for the angle, to create the pattern. If you still don’t believe me, I will just to the first 15 numbers;

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
    0 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 18.0 19.8 21.6 23.4 25.2 27.0

    Now you see what I mean? I can’t fit all 25 numbers for width, so that is why I stopped at 15. Now, you are probably saying “wow, that is awesome, but what angle do I have to measure?” Here;
    If you want an angle between a, c, you do
    a/2 = x
    45/x = y
    yb = z.

    if you want an angle between c, b, you do;
    c/2 = x
    45/x = y
    yb = z

    if you want to do the angle of b, a, you do;
    b/2= x
    45/x = y
    yb = z.

    That last formula to measure the angle of b, a, that equals 90. The angle would be a right-angle, and so it should; it is a right angle triangle. And get this; if you always multiply (y) by the first dividend, you always get 90.
    a/2 = x
    45/x = y
    y x a = z. (z) = 90.

    b/2 = x
    45/ x = y
    y x b = z. (z) = 90.

    For c, however, if c is a decimal, you will not get 90. You will get 89.9999 recurring.
    c/2 = x
    45/x = y
    y x c = z. (z) = 89.9999 recurring.

    Now how do you know what to divide by 2 and what to multiply by (y)? Well, one, you always multiply (y) by (b), because (b) is the base. And, two, always divide the measurement of the arm pointing northernmost of your angle, or, never use (b) as the dividend (you don’t have to anyway because you know what degree a right-angle is; it’s 90 degrees).
    Here is another example;
    You have a length of 39 and a width of 63. a = length. b = width.
    39^2 + 63^2 = (1521 + 3969 = 5490).
    sqrt(5490) = 74.094534211370814177212983848863 = c.
    74.094534211370814177212983848863 – 39 = 35.094534211370814177212983848863.
    35.094534211370814177212983848863 divided by 20 = 1.7547267105685407088606491924432.

    39/2 = 19.5
    45/19.5 = 2.3076923
    2.3076923 x 63 = 145.384615.

    The angle of arm a (39), and arm c (74.094534211370814177212983848863) has 145.384615 degrees.

    74.094534211370814177212983848863/2 = 37.047267105685407088606491924432 (oh look at that, the answer to (c – a)/b actually has those decimal numbers in it after two decimal places!)
    45/37.047267105685407088606491924432 = 1.2146644952628440423327486057153.
    1.2146644952628440423327486057153 x 63 = 76.523863201559174666963162160063.

    The angle of arm c (74.094534211370814177212983848863) and arm b (63) is 76.523863201559174666963162160063 degrees.

    63/2 = 31.5
    45/31.5 = 1.428571
    1.428571 x 63 = 90. Why is that? Well, we just did 63/2, and then did 45/(the answer to 63/2). Pursuant to maths, we used 2 as our divisor (factor), and 45 as our following dividend. Reversing the equation; 45 x 2, we get? 90.

    Now since that is 90, the angle between arm b (63) and arm a (39) is 90.

    Thanks!

    P.S. I am only thirteen years of age.

  19. George says:

    The computer does not accept extra spaces between words (or numbers), so the 1 – 15 pattern may be very confusing. So here, I have made it easier for you.
    _
    1
    1.8
    _
    2
    3.6
    _
    3
    5.4
    _
    4
    7.2
    _
    5
    9.0
    _
    6
    10.8
    _
    7
    12.6
    _
    8
    14.4
    _
    9
    16.2
    _
    10
    18.0
    _
    11
    19.8
    _
    12
    21.6
    _
    13
    23.4
    _
    14
    25.2
    _
    15
    27.0
    _

  20. George says:

    Plus, instead of measuring 3 inches then multiplying the distance between them by 20, you could measure 1.5 inches then multiply the distance by 10, or measure 0.75 inches and multiply the distance by 4.

    Thanks.

    My email is george21psa@gmail.com if you want to contact me.

  21. mathematics_are says:

    This is a cool and helpful tip, for those wondering how accurate it is:
    For a triangle with sides of length 3, 3,and x (after doing some trig) you will find that the relationship for the angle(lets call it T) between those two identical sides and the value x is:
    A = 2arcsin(x/6)
    If you instead use the approximation given in this article:
    A = 20x
    You’ll find that the smaller x is, the closer these two numbers are. However, you’ll notice that these two numbers are noticeably different when x is 4.243 (which should make A 90 degrees) , differing by just over 5 degrees. You should not use this approximation for any angles past 90 however as after this, as the error grows quite significantly(when you use the value 6 for x, you’ll see that the actual angle A should be 180 degrees but with the approximation, its only 120!)

  22. Pingback: How To Find A Missing Angle Without A Protractor | Binomial Probability

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