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History

It is speculated that the seemingly-arbitrary number $360$ used to indicate a full revolution in degrees was chosen because the Babylonians counted in base $60$, and $60 \times 6 = 360$, making $360$ a nice even multiple of the base, which would have the same appeal as even multiples of $10$ have to us.

Alternative 1

Gradians. Each quadrant is assigned a range of $100$ gradians. A full revolution is therefore $400$ gradians. This is more intuitive for users of a base-10 numerical system. At the time of writing, the Wikipedia article states:

Although attempts at a general introduction were made, the unit was only adopted in some countries and for specialized areas such as surveying, mining and geology.

Alternative 2

A full revolution could be $100$ units. The number $100$, which is $10 \times 10$, would have great aesthetic appeal in our number system.

Not only would this make arithmetic involving perpendicular and opposite angles easier, it would facilitate teaching the concept of angles to primary school children because of everybody’s familiarity with providing a quantity out of $100$ such as percentages, or school grades in many countries.

Practicality

Turkey switched from using an Arabic script to using a Latin script smoothly over a period of 4 years, and hasn’t looked back since. It could be argued that changing from degrees to another angle measure is a small change in comparison to changing a whole script.

Don’t we have radians?

Sure there’s also radians but we still use degrees a lot, in education for example. People who do not end up taking more advanced mathematics (most people) never learn what radians are. Also, most protractors work with degrees as opposed to radians.

Question

So, given the argument for a switch, is there a reason that we stick to using degrees?

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  • $\begingroup$ In Alternative 2, $60 \times 60 \neq 360$ $\endgroup$ – Noble Mushtak Jun 21 '16 at 19:00
  • $\begingroup$ @NobleMushtak Oops, fixed! $\endgroup$ – hb20007 Jun 21 '16 at 19:01
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    $\begingroup$ For any "serious" use, mathematicians use radians. So, degrees have been replaced. Degrees are convenient because 360 has a lot of factors. And 400 gradians in a circle is even more arbitrary in my mind than 360 degrees. $\endgroup$ – Doug M Jun 21 '16 at 19:02
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    $\begingroup$ Degrees have largely been decimalized already, minute and second are less and less used. $\endgroup$ – André Nicolas Jun 21 '16 at 19:08
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    $\begingroup$ Why weren't inches and ounces replaced with more intuitive measures? $\endgroup$ – Greg Martin Jun 21 '16 at 19:38
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History aside, you would like all the angles that commonly occur in elementary plane geometry (and are classically constructible and are rational multiples of $\pi$) to be integer numbers of degreess. Given that we need to put $90^\circ$, $60^\circ$ (in an equilateral triangle) and $45^\circ$ (in a right isosceles triangle) and $72^\circ$ (in a regular pentagon, which is constructible) to be integer numbers of degrees, we want to divide the circle into a number of degrees that is divisible by $60$. That rejects $400$.

In fact, if we also impose that a one degree angle difference should be something that can be discerned but not easily, the choices that make sense are that a circle is $300, 360, or 420$ degrees.

The choice of $360$ is somewhat historical and arbitrary among those.

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  • $\begingroup$ Possibly the most sensible answer among the many $\endgroup$ – Jikku Jose Apr 6 '20 at 5:48
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For high schoolers, degrees would definitely be easier than gradians. This is because $360$ is divisible by $2$, $3$, and $5$, so many angles of interest are integers in degrees while they are not in gradians: $$\frac{\pi}{3}=60^\circ=\frac{200}{3}^\text{g}$$ $$\frac{\pi}{4}=45^\circ=50^\text{g} \ \text{(OK, this works out for both.)}$$ $$\frac{\pi}{10}=18^\circ=20^\text{g} \ \text{(This works out, too; also, really only used in pentagons)}$$ $$\frac{\pi}{6}=30^\circ=\frac{100}{3}^\text{g}$$ $$\frac{\pi}{12}=15^\circ=\frac{100}{6}^\text{g} \ \text{(This is used a lot in pre-calc for half-angle identities.)}$$ Thus, even if gradians are useful in fields like surveying where angles are measured and used more numerically, in school, while angles are often given to us as an arbitrary decimal in trigonometry, we also often had to deal with these very specific angles, so it's really more helpful for those angles to be integers than for all of the decimals to be changed a little by one quadrant being $100^\text{g}$ instead of $90^\text{g}$.

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