How many degrees are there in a circle?

360, and the reason is Babylonian again - a base-sixty counting system, and a year of roughly 360 days, so the sun appeared to move about one degree round the sky each day. It survived because 360 has an extraordinary number of divisors: it splits evenly into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths, which no round number in base ten manages.

Mathematicians mostly do not use it. The natural unit is the radian, defined by wrapping the radius round the circumference, which gives a full circle of 2 pi radians - an awkward number until you get to calculus, at which point every formula involving angles becomes simpler and the degree version acquires stray conversion factors.

There have been attempts to reform it. The French Revolution, having decimalised weights, measures, money and the calendar, also decimalised the circle into 400 gradians, so a right angle was a neat 100. It failed like the ten-day week failed, though gradians survive in some surveying instruments and there is still a grad button on most scientific calculators that nobody presses.