Most of us have been wondering why we use 360 degrees for a full circle. Well, blame the Babylonians. Just kidding! The Babylonians used 60 as their base. While, we, use 10 as our base. So, what the Babylonians did is that they multiplied their base (60) by 6 (the number of sides in a hexagon). This system is called the sexagesimal system.
I would say that we use 360 because, it has a lot of advantages. Like for example, it's one of the few numbers that have many factors. These are all the factors:1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 . Having many factors makes 360 a very nice unit to divide a circle into equal parts. Also, we use it in calendars, and fractions. 360 is a very popular number.
This is an example: earth's revolution around the sun is approximately 360 days (Babylonians had 360 days a year).
So, I hope you understand a bit better of why we use 360. If I missed anything, or if you have any suggestions, fell free to comment. :)