A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

in which one

full rotation A turn is a unit of plane angle measurement equal to radians, 360 degrees or 400 gradians. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, etc. The closely related terms ''cycle'' and ''rev ...

is 360 degrees. It is not an

SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...

—the SI unit of angular measure is the radian—but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians.

History

Equilateral chord with length equal to radius

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. 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 and the

Babylonian calendar The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new lunar phase, crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed b ...

, 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. The earliest trigonometry, used by the

Babylonian astronomers Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia. Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations ...

and their

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

successors, was based on

chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ...

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 Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus,

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scienti ...

, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.

Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexand ...

used a simpler sexagesimal system dividing a circle into 60 parts. Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s,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. making it one of only 7 numbers such that no number less than twice as much has more divisors . Furthermore, it is divisible by every number from 1 to 10 except 7.Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10. This property has many useful applications, such as dividing the world into 24

time zone A time zone is an area which observes a uniform standard time for legal, commercial and social purposes. Time zones tend to follow the boundaries between countries and their subdivisions instead of strictly following longitude, because i ...

s, each of which is nominally 15° of

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

, to correlate with the established

24-hour The modern 24-hour clock, popularly referred to in the United States as military time, is the convention of timekeeping in which the day runs from midnight to midnight and is divided into 24 hours. This is indicated by the hours (and minutes) pas ...

day A day is the time period of a full rotation of the Earth with respect to the Sun. On average, this is 24 hours, 1440 minutes, or 86,400 seconds. In everyday life, the word "day" often refers to a solar day, which is the length between two so ...

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 Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

or for geographic coordinates (

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...

and

), degree measurements may be written using decimal degrees (''DD notation''); for example, 40.1875°. Alternatively, the traditional sexagesimal

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...

subdivisions can be used: one degree is divided into 60 ''minutes (of arc)'', and one minute into 60 ''seconds (of arc)''. Use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the '' arcminute'' and '' arcsecond'', are represented by a single prime (′) and double prime (″) respectively. For example, . Additional precision can be provided using decimal fractions of an arcsecond. Maritime charts are marked in degrees and decimal minutes to facilitate measurement; 1 minute of latitude is 1 nautical mile. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25). The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by al-Kashi 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

" (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. SI prefixes can also be applied as in, e.g., millidegree, microdegree, etc.

Alternative units

In most

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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 Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...

(360°) is equal to 2'' '' radians, so 180° is equal to radians, or equivalently, the degree is a mathematical constant: 1° = . The

(corresponding to a cycle or revolution) is used in

technology Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, scie ...

and

science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...

. One turn is equal to 360°. With the invention of the metric system, based on powers of ten, there was an attempt to replace degrees by decimal "degrees" in France and nearby countries,These new and decimal "degrees" must not be confused with decimal degrees. where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = gon). This was called or '' grad''. Due to confusion with the existing term ''grad(e)'' in some northern European countries (meaning a standard degree, of a turn), the new unit was called in German (whereas the "old" degree was referred to as ), likewise in Danish,

Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...

and Norwegian (also ''gradian''), and in Icelandic. To end the confusion, the name ''gon'' was later adopted for the new unit. Although this idea of metrification was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades () 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 to . It is approximately equal to one milliradian ( ). A mil measuring of a revolution originated in the

imperial Russian army The Imperial Russian Army (russian: Ру́сская импера́торская а́рмия, Romanization of Russian, tr. ) was the armed land force of the Russian Empire, active from around 1721 to the Russian Revolution of 1917. In the earl ...

, 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 Indirect fire is aiming and firing a projectile without relying on a direct line of sight between the gun and its target, as in the case of direct fire. Aiming is performed by calculating azimuth and inclination, and may include correcting ai ...

artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Notes

References

External links