The degree is a unit of plane angle equal to one three-hundred-and-sixtieth of a full turn. Its symbol is a small raised circle, and the abbreviation deg is used where that character is unavailable. It is not an SI unit but is accepted for use with the SI, and it remains the ordinary way of stating an angle outside pure mathematics.
The choice of 360 is Babylonian. Their sexagesimal arithmetic worked in sixties, and 360 is close to the number of days in a year, so the sun appears to advance roughly one degree along the ecliptic each day. The number is also unusually convenient: it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180, so most common fractions of a circle land on whole degrees.
Subdivision follows the same base. A degree splits into 60 minutes of arc and each minute into 60 seconds of arc, which is why latitude and longitude are written in degrees, minutes and seconds. Decimal degrees have largely replaced that notation in software, since a single number is easier to compute with, but navigation charts and astronomical catalogues keep the older form.
Everything from surveying and construction to camera lenses, screw threads and compass bearings is specified in degrees. A right angle is 90, a straight line 180, a full rotation 360, and those three facts carry most everyday geometry without any conversion at all.
The metric system tried to replace it and failed. Revolutionary France introduced the gradian, one hundredth of a right angle, so that a quarter turn was 100 units and a full turn 400. It survives in a few surveying instruments in continental Europe and in the gradian key still found on pocket calculators, but nothing else adopted it. Modern practice has instead moved away from minutes and seconds while keeping the degree itself: satellite navigation, mapping software and geographic databases store latitude and longitude as decimal degrees, so 51 degrees 30 minutes becomes 51.5. The two notations describe the same angle, and confusing one for the other misplaces a position by tens of kilometres.
One degree equals about 0.0174533 radians, 60 minutes of arc, or 3600 seconds of arc.
| Unit | Symbol | 1 Degree equals | 1 of these equals | Converter |
|---|---|---|---|---|
| Second of arc | arcsec | 3600 arcsec | 0.000277777777778 deg | Degrees to Seconds of arc |
| Minute of arc | arcmin | 60 arcmin | 0.0166666666667 deg | Degrees to Minutes of arc |
| Radian | rad | 0.0174532925199 rad | 57.2957795131 deg | Degrees to Radians |