As it has been replied here - on Wonder Quest (webarchive link):

The Sumerians watched the Sun, Moon, and the five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn), primarily for omens. They did not try to understand the motions physically. They did, however, notice the circular track of the Sun's annual path across the sky and knew that it took about 360 days to complete one year's circuit. Consequently, they divided the circular path into 360 degrees to track each day's passage of the Sun's whole journey. This probably happened about 2400 BC.

That's how we got a 360 degree circle. Around 1500 BC, Egyptians divided the day into 24 hours, though the hours varied with the seasons originally. Greek astronomers made the hours equal. About 300 to 100 BC, the Babylonians subdivided the hour into base-60 fractions: 60 minutes in an hour and 60 seconds in a minute. The base 60 of their number system lives on in our time and angle divisions.

An 100-degree circle makes sense for base 10 people like ourselves. But the base-60 Babylonians came up with 360 degrees and we cling to their ways-4,400 years later.