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Apologies if this question has some obvious answer. Why are degrees still taught as a measure for angles, to be replaced later by radians (probably confusing many people), rather than just starting with radians in the first place?

e.g. one could define the circumference of a circle to be $\tau$ or $2\pi$ and then immediately say "let's define the size of a central angle as $\tau$ times the proportion of the circumference that its arc subtends"...

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You could also propose to measure lengths not in Metres but Lightseconds. The answer is the same: it's not practical. –  Toscho Apr 10 at 20:50
To echo what @Toscho said (as well as some of the answers below), radians are really only interesting or useful for mathematics and physics. The wider world uses almost exclusively degrees, because they are much better suited to measuring angles in real-life situations. For example, I don't think I've ever heard anyone quote a longitude or latitude in radians, and both surveying and maritime navigation are based entirely on degrees. Angles are also used quite a lot in sports, architecture, visual arts, carpentry, manufacturing etc., and no one in any of these fields uses radians. –  Jim Belk Apr 11 at 17:27
@JimBelk "Better suited" is difficult. For small angles, degrees are better suited, because the numbers are low. For large angles, radians is better suited, but only if given as fractions of $\tau$. –  Toscho Apr 11 at 17:44
@Toscho - if a carpenter needs to communicate a 120 degree angle as a measurement of some part to another carpenter - radians still aren't better suited, although the angle is large. If some domain uses degree angles, then any radian measurements would/should be eliminated, no matter if the angle is small or large. –  Peteris Apr 11 at 22:41
I think to focus too much on degrees being somehow better for applications misses the point a bit, since actually in some areas a unit other than degrees is/was used. The key point to me regarding applications is that quite rarely one uses an angle bigger than a full turn. It is thus certainly inconvention (not a good choice) to use this as a base unit (often times). So one also has another unit (or other units), just like for any number of other things like distances. What the units used in applicationss seem to share is that the order of magnitude is convenient. –  quid Apr 12 at 6:18
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7 Answers 7

As others said, degrees are taught, since they are still used. So, the question becomes why are they still used.

To purely work with fractions would not be very convenient for various somewhat everyday things, since many people are more used to/better at operating with integers. So to really use $\tau$ and fractions thereof seems incovenient, and one somehow will want to rescale.

And, one could say, this is actually how degrees arise, one rescales by a factor of $360$. Now the question becomes, why $360$ and not something else, such as perhaps $100$.

The first answer is that some do rescale by something else, there is another unit called gradian where a right angle has $100$ gradian, which came up along with the metric system. There is (are in fact) also mil and a couple other things, see the Wikipedia page on angle.

The second answer is that $360$ is quite reasonable, more so than things more based on $10$, if one is interested in geometry.

To construct a hexagon is an extremely natural thing, so obviously a sixth of a turn is important and should be 'nice.' And a right angle being important too, this should also be 'nice.' So aready the scale should be divisible by $12$. Now, throw in the mix that also the constrcution of a regular pentagon is known since ancient times, and you are at multiples of $60$.

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Add that a year is a full circle, also around 360 days... –  vonbrand Apr 10 at 23:45
And throw in the constructability of a heptakaidekagon and you are at multiples of 1020. –  Toscho Apr 11 at 17:52
An analogy would be temperature units. Of course, you could give temperatures as multiples of the difference of waters fixpoints (DWF). You could then be happy about a temperature of 0.25 DWF and melt on a temperature of 0.45 DWF. –  Toscho Apr 11 at 17:55
@Toscho why stop there? We could start a movement to use 17179869180 parts ;-) More seriosuly, the 17-gon construction is due to Gauss, IIRC, so it is not known "since ancient times" (by contrast the 5-gon construction is known since antiquity). –  quid Apr 11 at 18:09
@quid Ah, I forgot, that "since ancient times" beats Lizard and Spock. –  Toscho Apr 11 at 19:07
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This is a good example for the Spiral approach:

Angle and degrees are introducted in late primary or early secondary. At that stage, students only know integers and proportions are too abstract.

Later on in mid-secondary, they learn about arc length and wedge area to be proportional to the angle. Depending on how well the students understand proportions and rescaling, one could introduce radians here as a proportional rescaling of degrees.

And finally in late-secondary, they learn about trigonometric functions, circle movements and oscillations, where radians has advantages over degrees. At this stage, they should learn to use both, because both units are just tools for other results. Here, you may redefine an angular measure.

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First, it’s a historically grown and widely used standard and as such it is as difficult to change as our weird time system, the decimal number system, QWERTY-based keyboards and English orthography. In particular, there are far too many people who only know the degree system and with whom children could not communicate, had they only been taught radians. Also note the failed attempt of introducing gradians.

Second, If one tried to introduce such a system, it would arguably be a good idea to introduce $π$ like any other unit and tell the students much later that it actually is a number. But even then, it’s not that advantageous, as the mostly used angles would be $π/4$, $π/3$ and so on, which is not that practical for most people. Of course one could introduce a new unit corresponding to $π/12$ or such, but that would be not be so different from degrees anymore.

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  • 360 has 24 divisors, more than any smaller number (this is called a highly composite number).
  • None of its prime factors is larger than 5 ("5-smooth").
  • It can be divided by every natural number between 1 and 10, with the exception of 7. No smaller number can.

Plus a couple more characteristics that make it very much suited for subdivision. Thus, many applications of "degrees" allow the use of integers instead of fractions, which most people are far more comfortable with.

Remember all the people working with angles all day that do not have an academic degree to do fancy maths: Carpenters, masons, seamstresses... they prefer a natural number over multiples or fractions of some funny constant any time.

By the way, that's also the reason why we subdivide time by 24 and 60 instead of the more "natural" 10, or "fractions of day": Ease of subdivision. Thankfully, our forbearers were a lazy lot. ;-)

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I'd say that your second point was a minus not a plus in that you'd want as many prime divisors as possible (except that you'd end up with an even bigger number so then it's not worth the gain). –  Loop Space Apr 11 at 15:34
@AndrewStacey: Having large prime factors means that you could end up with a rather large number that you cannot divide without getting a fraction. 360 being 5-smooth means that any subdivision of 360 larger than 5 can still be divided further. You wouldn't want to end up with, say, 37 -- 18.5 -- 9.25 -- 4.625 -- 2.3125 -- ... –  DevSolar Apr 11 at 17:27
That's what I meant by "not worth the gain". –  Loop Space Apr 11 at 17:49
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Degree measure is used in practical applications such as civil engineering, surveying, and astronomy. Other trigonometric conventions used in these applied branches are different from those in mathematics: for instance, angles are conventionally measured clockwise from North, rather than counterclockwise from the positive X axis.

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I don’t think that the orientation issue really plays into this. Also you forget about all the people far from academia using degrees. –  Wrzlprmft Apr 10 at 22:12
It doesn't, really, except to indicate that the use of degrees isn't the only difference between theoretical and applied mathematics. –  Confutus Apr 10 at 22:18
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I used to be firmly in the "Grr, what use are degrees?" camp until I actually started to use angles and then I switched to a "Why would anyone ever use radians?" attitude.

So in short, my current attitude is that if someone is likely to want to use actual angles, teach them degrees. If their interest is theoretical, then they might be able to cope with radians.

My use-case that switched my attitude was that of graphical programming. I frequently want to draw something on the screen rotated by some angle. I'm far more likely to want to rotate it by some fraction of a circle than anything else, so I want to work with proportions of a full circle. At this point, I could use degrees, gradians, or radians equally well.

The key, though, is in the accuracy of specification and storage of that angle. I want to know that if I rotate by a sixth of a circle and do it six times then I will rotate by a full circle. Now, for a sixth then the discrepancy is unlikely to be that great, but in fact I'll do this for much smaller angles. However I specify the angle, the resulting matrix is not going to be exact due to precision issues. So continually composing the matrix will eventually diverge from what it should be. Therefore my best strategy is to keep track of the angle and compute the matrix from the angle each time. But this won't work if my angle is not precise. So I want to be able to store the angle as precisely as I can. And if I store π or τ or some fraction thereof, it will only ever be an approximation. But if I store 10 or 20 or 30 then I can store it exactly and do exact calculations on it, thereby minimising the error in the final transformation.

Then why degrees instead of gradians or just proportions? The reason to discard just proportions is that most computer programs do not handle fractions exactly "out of the box" whereas they do handle integers exactly. So for the range of values we're actually interested in, it is more efficient to work with multiples of a fixed small angle than work with fractions of the whole. As for gradians, well, the Babylonians were right: 360 is a brilliant number of pieces to divide something into because there are so many factors that we can handle.

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Excel doesn't seem to handle degrees out of the box. When I want cosine of 75 degrees, for example, I have to type =cos(75*pi()/180). For all the trig functions I need to convert to radians. –  HopDavid Apr 11 at 13:11
@HopDavid Or you could use the RADIANS() function that converts from degrees to radians. –  Loop Space Apr 11 at 15:22
Thanks Andrew. If I had known that from the beginning, that's probably how I'd be doing it. However my conversion factor method is now an ingrained habit. And both methods seem to entail 9 keystrokes. –  HopDavid Apr 11 at 19:13
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Measuring in radians is useful when doing math, but it is terribly awful in the real life. It's not a chance that we still use a non-decimal measurement for degrees, since it is quite easy to divide in various numbers of parts.

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