In reality, mathematicians devide a circle into 400 slices, called radians. Every 90 degrees is equal to 100 radians. 100 is part of the base 10 measuring system, which is what the metric system is based off of. But we're not going to talk about the metric system, or radians here.
Think back for a moment to my last post, where I discussed the number of days in a year. It takes 365 days for the Earth to make a complete orbit around the sun. The Earth's orbit is a path that is very close to that of a circle.
The Ancients (not necessarily the same ones from my last post), while attempting to figure out circle mathematics, didn't have a very good time trying to figure stuff out by splitting the circle into 365 slices, but found that 360 gave them a lot more power. You could easily devide the circle in half, thirds, quarters, fourths, fifths, sixths, ninths, tenths, twelvths, and so on, and still retain a non-fraction number of degrees.
Since the Gods (of the time) were infallible, the only reason they could come up with that the Gods would split Earth into 365 parts (days) was because they wanted the people to have 5 days for holidays. So the Gods had split Earth's orbit into 360 degrees, plus 5 more for their rest/play/reward days. Obviously, the Gods knew about how easy math would be if a circle was devided into 360 degrees, and wanted the people to learn and come to understand this math, but they also wanted the people not to work every day... so this was the compromise.
If you think about this, it won't take you too long to realize that every day that passes, planet Earth moves one degree around the sun. After 360 degrees/days, you get a bonus five days, as decreed by the Gods, and so that the Earth can get back to the right position to start the next year.