A circle is 360 degrees because....
the Babylonians used a base 60 number system. It seems clear that degrees were devised by ancient astronomers who noticed that the sun moved one degree each day (about our fixed Earth) past the stars that appeared to be fixed to an external heavenly sphere. It took one month for the sun to move the 30 degrees from one sign of the Zodiac to the next.
Since there are really 365 days in a year (not 360) why isn't a degree defined to be 1/365 of a circle instead of the official 1/360? I have seen several explanations, most not very convincing. My current favorite explanation is that the number 360 is a compromise between the solar year of about 365.25 days and the lunar year (consisting of 12 months of 29.5 days each) of about 354.37 days. Of course the months fit well with the Zodiac, which played an important role in their astronomy.
Sorry if this wasn't really that helpful, I got this from a site and this was the best one I got.