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.
Links on Measurement System Analysis: