Why are degree angle-measurements taught?

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 made by the arc that it is subtended by"...

• You could also propose to measure lengths not in Metres but Lightseconds. The answer is the same: it's not practical. Apr 10 '14 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. Apr 11 '14 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 $\tau$. Apr 11 '14 at 17:44
• Radians are only preferable because they are the natural way of measuring angles. We also prefer to use natural logarithms because base e is the most natural choice. These mathematical factors however don't really apply to everyday use of the concepts. Apr 12 '14 at 10:11
• @Toscho Except specifying radians as a fraction of $\tau$ is precisely turns: i.e. $1\cdot\tau$ radians is exactly one turn. That is what I meant. If you said it is most useful to specify angles in radians as a fraction of $\tau$ you are saying it is most useful to specify angles as number of turns. The two are intertwined. Apr 12 '14 at 19:40

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 $60$.

• Add that a year is a full circle, also around 360 days... Apr 10 '14 at 23:45
• And throw in the constructability of a heptakaidekagon and you are at multiples of 1020. Apr 11 '14 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. Apr 11 '14 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 '14 at 18:09
• @quid Ah, I forgot, that "since ancient times" beats Lizard and Spock. Apr 11 '14 at 19:07
• 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. ;-)

• 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). Apr 11 '14 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 -- ... Apr 11 '14 at 17:27
• That's what I meant by "not worth the gain". Apr 11 '14 at 17:49

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.

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.

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.

• 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. Apr 11 '14 at 13:11
• @HopDavid Or you could use the RADIANS() function that converts from degrees to radians. Apr 11 '14 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. Apr 11 '14 at 19:13

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.

• I don’t think that the orientation issue really plays into this. Also you forget about all the people far from academia using degrees. Apr 10 '14 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. Apr 10 '14 at 22:18

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.

I would call this an issue of measurement granularity. Consider this blurb from the Six Sigma process site:

Establishing the adequacy of your measurement system using a measurement system analysis process is fundamental to measuring your own business process capability and meeting the needs of your customer (specifications). Take, for instance, cycle time measurements: It can be measured in seconds, minutes, hours, days, months, years and so on. There is an appropriate measurement scale for every customer need/specification, and it is the job of the quality professional to select the scale that is most appropriate.

Consider the case where a practitioner in the field is measuring angles with a protractor or similar instrument, based on radians, i.e., approximately just 6 units per complete circle. Obviously that would be too imprecise, so you'd need subdivisions: perhaps the protractor is marked in hundredths of a radian? So then almost every measurement one makes would have to document some decimal places (e.g., 1.57 radians).

Difficulties of this approach: It's a bit inefficient to track a "point" for every single measurement. Important angles cannot be exactly expressed in this system -- 1.57 radians is only approximately a right angle, but you can't be any more precise with this level of granularity. As a result, it's also a bit difficult to calibrate or manufacture the protractor (granted that a full circle is actually an irrational number, not constructable by compass-and-ruler geometry, etc.)

So for convenience sake, it's reasonable to rescale the measurement to a finer granularity, such that practitioners are always dealing with whole numbers -- hopefully neither too big nor too small (see the example above for different options used for time measurements). One option would be to simply scale by a factor of 100 for the protractor in hundredths of a radian -- in fact, that's exactly analogous to the utility of using units of "percent" to express ratios that are usually smaller than one. Call this "centiradians" if you like: approximately 628 per circle.

But the remaining problem is that this level of granularity on the protractor is still only approximate for a full circle, half circle, right angle, etc. So better options would be to arbitrarily define the circle as a convenient whole number on the order of a few hundred. One option is 400 (gradians); as others point out, 360 (degrees) is nice for having lots of divisors (including 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.) -- that is, many important angles will be a whole-number of degrees. Thus, no fractions or decimals are needed in most cases, important angles can be exactly expressed, and it's about the most efficient measurement scale to document and communicate the angles that most people will be dealing with.