I think so. Can we now agree that your statement that only a circle has 360 degrees was either in error, or could be interpreted as being in error?
Nope. Sorry. And I'm gonna have to ramble on a bit here.....
Firstly, the term
degree is derived from the
Babylonian base 60 numerical system. Hours and minutes are similarly divided into 60s (of course, there are minutes of time and minutes of angle—there are 60 minutes in a degree, and, similarly, there are seconds of time and seconds of degree—there are 60 seconds in a minute, 3,600 in a degree).
Rather than degrees though,
more important is the radian. There are precisely 2pi radians in a circle, where pi is of course the ratio of the circumference to its diameter
And to answer your next logical question as to why the radian is defined as 2pi in a circle—considering that this number is not even an integer...
The reason is that the radian satisfies certain properties, such that trigonometry and calculus are made easier when you let angles be measured in radians. The underlying idea is that the circumference of a circle of radius 1 is 2pi. So in a circle of radius 1, one radian subtends an arc length of exactly 1.
This makes measuring arc length equivalent to measuring angle. Similarly, in a circle of radius 1, pi radians (which, by the way, equals 180º), subtends an arc length of pi.
I hope this is not too confusing (but probably is?) but it's just that mathematicians and astronomers usually define arcs of circles in radians rather than degrees.
Check this out: