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| − | [[Image:Degree diagram.svg|thumb|one degree]]
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| − | :''This article describes the unit of angle. For other meanings, see [[degree]].''
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| − | A '''degree''' (in full, a '''degree of arc''', '''arc degree''', or '''arcdegree'''), usually denoted by '''°''' (the [[degree symbol]]), is a measurement of [[plane (mathematics)|plane]] [[angle]], representing <sup>1</sup>⁄<sub>360</sub> of a full rotation; one degree is equivalent to π/180 [[radian]]s. When that angle is with respect to a reference [[meridian (geography)|meridian]], it indicates a location along a [[great circle]] of a [[sphere]], such as Earth (see [[Geographic coordinate system]]), [[Mars]], or the [[celestial sphere]].<ref>Beckmann P. (1976) ''A History of Pi'', St. Martin's Griffin. ISBN 0-312-38185-9
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| − | </ref>
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| − | ==History==
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| − | [[Image:Equilateral chord.svg|thumb||right|A circle with an equilateral [[Chord (geometry)]] (red). One sixtieth of this arc is a degree. Six such chords complete the circle]]
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| − | The selection of [[360 (number)|360]] as the number of degrees (''i.e.,'' smallest practical sub-arcs) in a circle was probably based on the fact that 360 is approximately the number of days in a year. Its use is often said to originate from the methods of the ancient [[Babylonian]]s.<ref>[http://mathworld.wolfram.com/Degree.html Degree], MathWorld</ref> Ancient [[astronomers]] noticed that the stars in the sky, which circle the [[celestial pole]] every day, seem to advance in that circle by approximately one-360th of a circle, ''i.e.,'' one degree, each day. (Primitive [[calendar]]s, such as the [[Persian calendar|Persian Calendar]], used 360 days for a year.) Its application to measuring angles in [[geometry]] can possibly be traced to [[Thales]] who popularized geometry among the [[Greeks]] and lived in Anatolia (modern western [[Turkey]]) among people who had dealings with [[Egypt]] and Babylon.
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| − | The [[history of trigonometry|earliest trigonometry]], used by the [[Babylonian astronomy|Babylonian astronomers]] and their [[Greek astronomy|Greek]] successors, was based on [[chord]]s 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; while six such chords completed the full circle.
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| − | Another motivation for choosing the number 360 is that it is readily divisible: 360 has 24 [[divisor]]s (including 1 and 360), including every number from 1 to 10 except 7. For the number of degrees in a circle to be divisible by every number from 1 to 10, there would need to be 2520 degrees in a circle, which is a much less convenient number.
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| − | :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.
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| − | === India ===
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| − | The division of the circle into 360 parts also occurred in ancient [[India]], as evidenced in the [[Rig Veda]]:
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| − | :Twelve spokes, one wheel, navels three.
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| − | :Who can comprehend this?
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| − | :On it are placed together
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| − | :three hundred and sixty like pegs.
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| − | :They shake not in the least.
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| − | :(Dirghatama, Rig Veda 1.164.48)
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| − | ==Subdivisions==
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| − | 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 [[latitude]]s and [[longitude]]s on the Earth, degree measurements may be written with [[decimal]] places, but the traditional [[sexagesimal]] [[Units of measurement|unit]] subdivision is commonly seen. One degree is divided into 60 ''minutes (of arc)'', and one minute into 60 ''seconds (of arc)''. These units, also called the ''[[arcminute]]'' and ''[[arcsecond]]'', are respectively represented as a single and double [[prime (symbol)|prime]], or if necessary by a single and double quotation mark: for example, 40.1875° = 40° 11′ 15″ (or 40° 11' 15").
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| − | If still more accuracy is required, decimal divisions of the second are normally used, rather than ''thirds'' of <sup>1</sup>⁄<sub>60</sub> second, ''fourths'' of <sup>1</sup>⁄<sub>60</sub> of a third, and so on. These (rarely used) subdivisions were noted{{Fact|date=January 2008}} by writing the [[Roman numeral]] for the number of sixtieths in superscript: 1<sup>I</sup> for a "prime" (minute of arc), 1<sup>II</sup> for a second, 1<sup>III</sup> for a third, 1<sup>IV</sup> for a fourth, etc. Hence the modern symbols for the minute and second of arc.
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| − | ==Alternative units==
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| − | :''See also: [[Angle#Measuring angles| Measuring angles]].''
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| − | In most [[mathematics|mathematical]] work beyond practical geometry, angles are typically measured in [[radian]]s rather than degrees. This is for a variety of reasons; for example, the [[trigonometric function]]s 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 circle (360°) is equal to 2''[[Pi|π]]'' radians, so 180° is equal to π radians, or equivalently, the degree is a [[mathematical constant]] ° = <sup>''π''</sup>⁄<sub>180</sub>.
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| − | With the invention of the [[metric system]], based on powers of ten, there was an attempt to define a "decimal degree" ('''[[grad (angle)|grad]]''' or '''gon'''), so that the number of decimal degrees in a right angle would be 100 ''gon'', and there would be 400 ''gon'' in a circle. Although this idea did not gain much momentum, most scientific [[calculator]]s used to support it. {{Fact|date=July 2008}}
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| − | An [[angular mil]] which is most used in military applications has at least three specific variants.
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| − | In computer games which depict a three-dimensional virtual world, the need for very fast computations resulted in the adoption of a binary, 256 degree system. In this system, a right angle is 64 degrees, angles can be represented in a single byte, and all trigonometric functions are implemented as small lookup tables. These units are sometimes called "binary radians" ("brads") or "binary degrees".{{Fact|date=September 2007}}
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| − | ==See also==
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| − | * [[Angle]]
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| − | * [[Compass]]
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| − | * [[Radian]]
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| − | == Notes ==
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| − | <references/>
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| − | ==References==
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| − | ==External links==
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| − | * [http://www.mathopenref.com/degrees.html Degrees as an angle measure], with interactive animation
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| − | * [http://mathworld.wolfram.com/Degree.html Degree] at [[MathWorld]]
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| − | [[Category:Physical sciences]]
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| − | [[Category:Mathematics]]
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| − | {{credit|249952671}}
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