Adegree (in full, adegree of arc,arc degree, orarcdegree), usually denoted by° (thedegree symbol), is a unit of measurement of aplaneangle in which onefull rotation is assigned the value of 360 degrees.[4]
It is not anSI unit—the SI unit of angular measure is theradian—but it is mentioned in theSI brochure as anaccepted unit.[5] Because a full rotation equals 2π radians, one degree is equivalent toπ/180 radians.
A circle with an equilateralchord (red). One sixtieth of this arc is a degree. Six such chords complete the circle.[6]
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. Ancientastronomers noticed that the sun, which follows through theecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancientcalendars, such as thePersian calendar and theBabylonian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use ofsexagesimal numbers.[4]
Another theory is that the Babylonians subdivided the circle using the angle of anequilateral triangle as the basic unit, and further subdivided the latter into 60 parts following their sexagesimal numeric system.[7][8] Theearliest trigonometry, used by theBabylonian astronomers and theirGreek successors, was based onchords 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.
Another motivation for choosing the number 360 may have been that it isreadily divisible: 360 has 24divisors,[note 1] making it one of only 7 numbers such that no number less than twice as much has more divisors (sequenceA02182 in theOEIS).[11] 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 24time zones, each of which is nominally 15° oflongitude, to correlate with the established24-hourday 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.
For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as inastronomy or forgeographic coordinates (latitude andlongitude), degree measurements may be written usingdecimal degrees (DD notation); for example, 40.1875°.
Alternatively, the traditionalsexagesimalunit subdivisions can be used: one degree is divided into 60minutes (of arc), and one minute into 60seconds (of arc). Use of degrees-minutes-seconds is also calledDMS notation.[12] These subdivisions, also called thearcminute andarcsecond, are represented by asingle prime (′) anddouble prime (″) respectively. For example,40.1875° = 40° 11′ 15″. 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 1nautical mile. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25).[13]
The older system ofthirds,fourths, etc., which continues the sexagesimal unit subdivision, was used byal-Kashi[citation needed] and other ancient astronomers, but is rarely used today. These subdivisions were denoted by writing theRoman numeral for the number of sixtieths in superscript: 1I for a "prime" (minute of arc), 1II for asecond, 1III for athird, 1IV for afourth, etc.[14] Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system.[15]
SI prefixes can also be applied as in, e.g.,millidegree,microdegree, etc.
In mostmathematical work beyond practical geometry, angles are typically measured inradians rather than degrees. This is for a variety of reasons; for example, thetrigonometric 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 completeturn (360°) is equal to 2π radians, so 180° is equal toπ radians, or equivalently, the degree is amathematical constant: 1° =π⁄180.
Oneturn (corresponding to a cycle or revolution) is equal to 360°.
With the invention of themetric system, based on powers of ten, there was an attempt to replace degrees by decimal "degrees" in France and nearby countries,[note 3] where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° =10⁄9 gon). This was calledgrade (nouveau) orgrad. Due to confusion with the existing termgrad(e) in some northern European countries (meaning a standard degree,1/360 of a turn), the new unit was calledNeugrad inGerman (whereas the "old" degree was referred to asAltgrad), likewisenygrad inDanish,Swedish andNorwegian (alsogradian), andnýgráða inIcelandic. To end the confusion, the namegon 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 manyscientific calculators support them. Decigrades (1⁄4,000) were used with French artillery sights in World War I.
^Chang, Kang-Tsung (2019).Introduction to Geographic Information Systems (9th ed.). New York: McGraw-Hill Education. p. 24.ISBN978-1-259-92964-9.LCCN2017049567.
