Degree (angle)
Degree  

Unit system  NonSI accepted unit 
Unit of  Angle 
Symbol  °^{[1]}^{[2]} or deg^{[3]} 
Unit conversions  
1 °^{[1]}^{[2]} in ...  ... is equal to ... 
turns  1/360 
radians  π/180 
gons  10/9^{g} 
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing ^{1}⁄_{360} of a full rotation. It is not an SI unit, as the SI unit for angles is radian, but it is mentioned in the SI brochure as an accepted unit.^{[4]} Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.
Contents
 History 1
 Subdivisions 2
 Alternative units 3
 See also 4
 Notes 5
 References 6
 External links 7
History
The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year.^{[5]} Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.
Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.^{[6]}^{[7]} The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree.
Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically.^{[8]}^{[9]} Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.^{[10]} Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.
The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda:^{[11]}
Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,^{[note 1]} making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in OEIS).^{[12]}^{[13]} Furthermore, it is divisible by every number from 1 to 10 except 7.^{[note 2]} This property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24hour day convention.
Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere and that it was rounded to 360 for some of the mathematical reasons cited above.
Subdivisions
For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates (latitude and longitude), degree measurements may be written using decimal degrees. For example, 40.1875°, with the degree symbol behind the decimals.
Alternatively, the traditional sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes (of arc), and one minute into 60 seconds (of arc). Use of degreesminutesseconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are respectively represented as a single and double prime. For example, 40.1875° = 40° 11′ 15″, or using quotation mark characters as 40° 11' 15". Additional precision can be provided using decimals for the arcseconds component.
The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by alKashi and other ancient astronomers, but is rarely used today. These subdivisions were denoted by writing the Roman numeral for the number of sixtieths in superscript: 1^{I} for a "prime" (minute of arc), 1^{II} for a second, 1^{III} for a third, 1^{IV} for a fourth, etc. Hence the modern symbols for the minute and second of arc, and the word "second" also refer to this system.
Alternative units
In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant: 1° = ^{π}⁄_{180}.
The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. 1 turn = 360°.
With the invention of the metric system, based on powers of ten, there was an attempt to replace degrees by decimal "degrees"^{[note 3]} called grad or gon, where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = ^{10}⁄_{9} gon). Although that idea was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades ^{1}⁄_{4,000} were used with French artillery sights in World War I.
An angular mil, which is most used in military applications, has at least three specific variants, ranging from ^{1}⁄_{6,400} to ^{1}⁄_{6,000}, each approximately equal to one milliradian. However, ^{1}⁄_{6,000} used by the Russian Army originated in Imperial Russia, where an equilateral chord was divided into tenths to give a circle of 600 units (this may be seen on a lining plane, an early device for aiming indirect fire artillery, dating from about 1900, in the St Petersberg Museum of Artillery).
Turns  Radians  Degrees  Gradians (Gons) 

0  0  0°  0^{g} 
1/24  π/12  15°  16 2/3^{g} 
1/12  π/6  30°  33 1/3^{g} 
1/10  π/5  36°  40^{g} 
1/8  π/4  45°  50^{g} 
1/2π  1  ca. 57.3°  ca. 63.7^{g} 
1/6  π/3  60°  66 2/3^{g} 
1/5  2π/5  72°  80^{g} 
1/4  π/2  90°  100^{g} 
1/3  2π/3  120°  133 1/3^{g} 
2/5  4π/5  144°  160^{g} 
1/2  π  180°  200^{g} 
3/4  3π/2  270°  300^{g} 
1  2π  360°  400^{g} 
See also
 Compass
 Geographic coordinate system
 Gradian
 Meridian arc
 Square degree
 Square minute
 Square second
 Steradian
Notes
 ^ The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
 ^ Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10.
 ^ These new and decimal "degrees" must not be confused with decimal degrees.
References
 ^ ^{a} ^{b} HP 48G Series – User's Guide (UG) (8 ed.).
 ^ ^{a} ^{b} HP 50g graphing calculator user's guide (UG) (1 ed.).
 ^ HP Prime Graphing Calculator User Guide (UG) (PDF) (1 ed.).
 ^ Bureau International des Poid et Mesures (2006). "The International System of Units (SI)" (8 ed.).
 ^ "Degree". MathWorld.
 ^
 ^
 ^ Rawlins, Dennis. "On Aristarchus". DIO  The International Journal of Scientific History.
 ^
 ^ "2 (Footnote 24)". Aristarchos Unbound: Ancient Vision / The Hellenistic Heliocentrists' Colossal UniverseScale / Historians' Colossal Inversion of Great & Phony Ancients / HistoryofAstronomy and the Moon in Retrograde! (PDF). DIO  The International Journal of Scientific History 14. March 2008. p. 19.
 ^
 ^ Brefeld, Werner. "Divisibility highly composite numbers".
 ^ Brefeld, Werner (2015). (not defined).
External links
 "Degrees as an angle measure"., with interactive animation
 "Degree". at MathWorld
 Gray, Meghan; Merrifield, Michael; Moriarty, Philip (2009). "° Degree of Angle". Sixty Symbols.
