Q : Why does a circle have 360 degrees?

A : We delve back to arrive at a probable answer. A line of ancient peoples (Sumerians, Akkadians, and Babylonians) who lived in Mesopotamia (now southern Iraq) invented writing, observed the skies, and invented a 360-degree circle to describe their findings. About 3000 BC, the Sumerians invented writing. They also had a calendar, dating from 2400 BC, that divided the year into 12 months of 30 days each, that is, 360 days.

[Corel] Ancient Babylonians may have invented the sundial

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. 

-- http://www.wonderquest.com/circle.htm