degree
Angleexact by definition
The degree is exactly 1/360 of a full turn. In radians that is \(\pi/180\), so the definition is exact even though the decimal value 0.01745329... is irrational and must be rounded when written out. It is not an SI unit but is accepted for use with SI.
| 1 ° | 1 ° |
The 360-part circle is inherited from Babylonian astronomy and its base-60 arithmetic, which also gave us 60 minutes to the degree and 60 seconds to the minute. The number survives because 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more, making common fractions of a circle land on whole degrees.
About the degree
Nothing in geometry requires 360. What recommends it is divisibility: 360 has 24 divisors, so a third of a circle is 120°, an eighth is 45°, a fifth is 72°, and all of them are integers. Try that with 100 and two of the three fail. The choice traces to Babylonian astronomy, where a base-60 number system met a year of roughly 360 days, and the Sun's apparent motion of about one degree per day made the correspondence natural. The subdivisions carry the same fingerprint: pars minuta prima gave the minute and pars minuta secunda gave the second, still 1/60 and 1/3600 of a degree today.
Attempts to replace it have failed twice. Revolutionary France pushed the grad, with 100 to a right angle, and it survives only in some European surveying. Radians won the mathematics but never took the drawing office, the compass rose or the protractor. The practical reason is that degrees are the right size for human work: one degree is a visible but small angle, roughly twice the width of the full Moon in the sky, and a whole-degree bearing is precise enough for navigation without needing decimals. Where more precision is needed, surveying uses degrees-minutes-seconds and astronomy uses arcseconds, both still inside the same Babylonian scheme.