That one rotation equals 360 degrees is just a convention, There is nothing partcular about 360 except that it can be divided by many numbers.
Another, and a more fruitful, approach to measuring angles is to use the length of the arc cut out by the angle on the perimeter of the unit-circle.
Since the circumference of the unit-circle is 2π this corresponds to 360 degree
π RADIANS then equal 180degrees and hence 1 radian = 180/π degrees.
It follows that one revolution corresponds to 2π such radians.
This reasoning can be expanded to arbitrary circles having radius r.
Their circumference being 2πr which means 2π radians proving that the angle is equal to 2π radians in this case as well.
To see an attempt to square the circle click on the figure below. Courtesy of Wiking Björkman (my grandfather):
To listen to an 1981 interview (Radio Kronoberg) with him click below
11 Spår 11_(new).mp3