Rankine.
Am I the only person here who wonders what the heck they searched to come up with this thread in the first place? 
Okay, why is that word in The Urban Dictionary? Shouldn’t that be just for real and made-up slang?
If you start questioning the ascholaricity of UD, you will end up finding terrifying holes in the universe. Do not do that. Just accept that someone fucked up, and move on. We do not need terrifying holes in the universe, because the lovecraft will come crawling out of one. I am not ready to be eaten by some mirthless elder god just yet.
The zombie-bump’s format reminds me of a quora question, suggesting a possible avenue for the reanimation.
As to the original question, “degrees” is a measurement of the angle formed by two lines. As such, it’s an inherently two-dimensional metric. If you want to measure a three-dimensional object like a sphere, you have to “upscale” your measurement to square degrees.
Similarly:
“inches” : a measure of the length of a line and is only good for one dimension.
“square inches” : a measure of the size of an area and is only good for two dimensions.
“cubic inches” : a measure of the volume of a space and is only good for three dimensions.
“Square” degrees sounds not quite right. If you dice a sphere through the center, you end up with 8 quadrants that have a semi-tetrahedral shape. The surface apices are all 90°, but they form a triangle. Any vector drawn from the center can be concisely described with two terms relative to any two of the three arc-sides of its quadrant – sort a, umm, to coin a term, “triangulation”.
That’s proving the point that square degrees are confusing and no one ever uses them, while steradians are an SI unit and everybody knows the surface area of a sphere is 4πr[sup]2[/sup].
I have a really good circle that has 370 degrees. It’s 10 degrees better than a regular circle.
I wonder what became of 3_14159265358979323846?
Is he still alive?
Has he evolved into a more rounded person?
We need more numeric posters.
If only he could give us some kind of sine.
We need one of his friends to come back, bring 3_14159265358979323846.
It’s not even quite wrong. 
But given the original statement, “360 degrees in a circle”, it’s probably the best you can do. Saying a circle has 360 degrees is only barely a measurement (as opposed to just an arbitrary definition of “degrees”). It tells you nothing about the size of the circle or anything else, it just suggests that from a point at the center of the circle, you could turn 360 degrees and you’d still see the circle. As opposed, I guess, to a circle-shape with a wedge cut out of it (i.e. a Pac-Man shape) where you might say “this has only 330 degrees in it”, i.e. a 30-degree wedge is missing.
It’s not correct to say that nobody ever uses square degrees. Astronomers do. Of course, we’re usually looking at a small enough patch of sky that the spherical curvature is negligible.
And I bet small enough patch that SI prefixes to radians would be less elegant and non-rational with compounding losses of precision.
In computers this a problem 0.1 = 1/10, and 10 is not a power of two. There will be an representation error without tricks which produces an irrational number and it may not symmetric for equivalence tests.
1 min arc = 1/21,600 of 1 turn = pi/10800 radians
1 arcsecond = 1/60 arcminute = 1/3600 degree = 1/1,296,000 turn = pi/648000 radians.
Probably not as big of an issue if you are calculating watts per steradian for RF or lasers but I can see why Astronomers would like the option of staying with rational numbers.
I think we all agree with 3_141592653589…'s answer, but the number of “degree confluences” may also have interest:
Of these 64,442 confluence points, 21,543 are on land, 38,409 on water, and 4,490 on the Antarctic and Arctic ice. There are 52 confluence points in California; the three in the Bay Area are near Santa Cruz, Concord, and the Point Reyes lighthouse. There’s one near Cinnaminson, Philadelphia. The confluence point nearest to me is a few miles south of an American-owned spaghetti sauce factory. Visit the Degree Confluence Project to help document the confluence point near you.
What does this mean? What’s the 2-D analogy?
rat avatar, rationality isn’t really an issue. Sure, something that’s a rational number of square degrees will be an irrational number of steradians, but the reverse is also true: Something that’s a rational number of steradians will be an irrational number of square degrees. And most contexts where it’d come up would be equally likely to be irrational in either units: That is to say, almost guaranteed, and so you just round it to whatever number of digits is convenient anyway.
Seriously, is there a hypersphere equivalent to radians, that defines the volume of a sphere in terms of the surface of a hypersphere?
Looking at the blue line on this graph you see 2π radians at 2D, 4π steradians at 3D, 2π[sup]2[/sup] at 4D, all the way up to π[sup]6[/sup]/60 at 12D.
I should have clarified that sticking with SI and prefixes is the issue.
The milliradian, mil, or mrad works fine in optics, but that style of prefixing would lead to issues going with a pure SI prefix model…
Consider parallax:
d = 1/p
And then the implications for the parsec, which would require trig to avoid FP loss of precision on small numbers or fighting with representation errors in binary.
Sure the above has some errors due to using sine small-angle approximation, but that error is 0.01ppm below one degree and gets smaller. That said I am focused on the limitations of computers.
A couple of other quick FP issues that make it nice to have sub units that are still rational.
Subtracting similar numbers leads to a loss of precision, Which due to the cumultive effect caused this Patriot missile failure.
tan(π/2) will not return infinity or an error.
Addition and Multiplication will not be reliably associative so: (foo + bar) + baz != foo + ( bar + baz ). This the big one I would worry about if we were to move everything to SI prefixes for magnitude to get rid of Deg/Min/Sec.
Obviously we use floats for lots of numbers, but there is a lot of value in avoiding irrational numbers when possible.
from decimal import Decimal
Decimal.from_float(0.1)
Out[25]: Decimal(‘0.1000000000000000055511151231257827021181583404541015625’)
Decimal.from_float(0.01)
Out[26]: Decimal(‘0.01000000000000000020816681711721685132943093776702880859375’)
Decimal.from_float(0.001)
Out[27]: Decimal(‘0.001000000000000000020816681711721685132943093776702880859375’)
Decimal.from_float(0.0001)
Out[28]: Decimal(‘0.000100000000000000004792173602385929598312941379845142364501953125’)
Decimal.from_float(0.0000001)
Out[29]: Decimal(‘9.99999999999999954748111825886258685613938723690807819366455078125E-8’)
The point is that degrees, min, sec will not suffer this issue as much if you can stay rational, With arc seconds you can pi until you need it too use it.
The loss of precision you’re describing will never be relevant for actual astronomy, and even if it were, the solution would be to use higher precision floats.