Hi everyone, There are many theories behind this and all are plausible answers to this question.
1The Babylonians used to use a chord of length equal to the radius in order to measure a circle. Such a chord makes an angle of 60 degrees at the centre. Since the Babylonians used a base 60 number system, they subdivided this 60-degree angle into 60 equal parts giving birth to 1 degree as the lowest measurable angle. Since a circle can have 6 such chords, the complete angle became 360 degrees.
2 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 favourite 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. Another nifty explanation, but which I've never seen confirmed by a reliable source: an ancient Persian calendar showing a year of 360 days has been discovered, so maybe the year at that time was 5 days shorter than our year!
3Lastly, if you were to take the radius of a circle and make an equilateral triangle with it (identical sides and angles) you'd find that six of them fit beautifully into the circle. In the past, people used the sexagesimal system which has a base of 60 (we use the decimal system now, that has a base of 10). It's thought that each of these triangles were arbitrarily given the value of 60. And 6 x 60 = 360.
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