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There are a number of theories: the ancient Babylonians counted in the sexagesimal system (counting in 60s). This was extended to 6*60 = 360, a number which was close to the number of days in the year. Also, 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20 etc which is convenient.

In any case, pure mathematicians do not measure angles in degrees - they use radians.

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Q: Why do you measure angles in 360 degrees?
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