Editing Angle (section)

===Units===
[[Image:Angle radian.svg|right|thumb|150 px|Definition of 1&nbsp;radian]]

Throughout history, angles have been [[measure (mathematics)|measured]] in various [[unit (measurement)|units]]. These are known as '''angular units''', with the most contemporary units being the [[degree (angle)|degree]] ( ° ), the [[radian]] (rad), and the [[gradian]] (grad), though many others have been used throughout [[History of Mathematics|history]].<ref>{{Cite web|title=angular unit|url=https://www.thefreedictionary.com/angular+unit|access-date=2020-08-31|website=TheFreeDictionary.com}}</ref> 

In the [[International System of Quantities]], an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in [[dimensional analysis]].

The following table lists some units used to represent angles.

{|class = "wikitable"
!Name !!Number in one turn!!In degrees !!Description
|-
|[[radian]]||{{math|2''π''}}||≈57°17′45″||The ''radian'' is determined by the circumference of a circle that is equal in length to the radius of the circle (''n''&nbsp;=&nbsp;2{{pi}}&nbsp;=&nbsp;6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is ''rad''. One turn is 2{{math|π}}&nbsp;radians, and one radian is {{sfrac|180°|{{pi}}}}, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit ''rad'' being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the [[trigonometric function]]s display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the [[SI]].
|-
|[[degree (angle)|degree]] ||360 ||1°|| The ''degree'', denoted by a small superscript circle (°), is 1/360 of a turn, so one ''turn'' is 360°. One advantage of this old [[sexagesimal]] subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the [[Minute and second of arc|"minute" and "second"]] sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for [[Geographic coordinate system|geographical coordinates]] and in [[astronomy]] and [[ballistics]] (''n''&nbsp;=&nbsp;360) 
|-
| [[arcminute]]||21,600 ||0°1′|| The ''minute of arc'' (or ''MOA'', ''arcminute'', or just ''minute'') is {{sfrac|60}} of a degree = {{sfrac|21,600}} turn. It is denoted by a single prime (&nbsp;′&nbsp;). For example, 3°&nbsp;30′ is equal to 3&nbsp;×&nbsp;60&nbsp;+&nbsp;30&nbsp;=&nbsp;210 minutes or 3&nbsp;+&nbsp;{{sfrac|30|60}} = 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3°&nbsp;5.72′ = 3&nbsp;+&nbsp;{{sfrac|5.72|60}} degrees. A [[nautical mile]] was historically defined as an arcminute along a [[great circle]] of the Earth. (''n''&nbsp;=&nbsp;21,600).  
|-
| [[arcsecond]]||1,296,000 ||0°0′1″||The ''second of arc'' (or ''arcsecond'', or just ''second'') is {{sfrac|60}} of a minute of arc and {{sfrac|3600}} of a degree (''n''&nbsp;=&nbsp;1,296,000). It is denoted by a double prime (&nbsp;″&nbsp;). For example, 3°&nbsp;7′&nbsp;30″ is equal to 3 + {{sfrac|7|60}} + {{sfrac|30|3600}} degrees, or 3.125&nbsp;degrees. The arcsecond is the angle used to measure a [[parsec]]
|-
| [[grad (angle)|grad]]||400 ||0°54′ || The ''grad'', also called ''grade'', ''[[gradian]]'', or ''gon''. It is a decimal subunit of the quadrant. A right angle is 100 grads. A [[kilometre]] was historically defined as a [[centi]]-grad of arc along a [[meridian (geography)|meridian]] of the Earth, so the kilometer is the decimal analog to the [[sexagesimal]] [[nautical mile]] (''n''&nbsp;=&nbsp;400). The grad is used mostly in [[triangulation (surveying)|triangulation]] and continental [[surveying]].
|-
|[[turn (geometry)|turn]]||1||360° || The ''turn'' is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2{{pi}} or [[Turn_(angle)#Proposals_for_a_single_letter_to_represent_2π|{{tau}} (tau)]] radians.
|-
| [[hour angle]] || 24 || 15° || The astronomical ''hour angle'' is {{sfrac|24}}&nbsp;turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called ''minute of time'' and ''second of time''. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1&nbsp;hour = 15° = {{sfrac|{{pi}}|12}}&nbsp;rad = {{sfrac|6}}&nbsp;quad = {{sfrac|24}}&nbsp;turn = {{sfrac|16|2|3}}&nbsp;grad.
|-
| [[Points of the compass|(compass) point]] || 32 || 11°15′ || The ''point'' or ''wind'', used in [[navigation]], is {{sfrac|32}} of a turn. 1&nbsp;point = {{sfrac|8}} of a right angle = 11.25° = 12.5&nbsp;grad. Each point is subdivided into four quarter points, so one turn equals 128.
|-
| [[milliradian]] || {{math|2000''π''}} || ≈0.057° || The true milliradian is defined as a thousandth of a radian, which means that a rotation of one [[Turn (geometry)|turn]] would equal exactly 2000π&nbsp;mrad (or approximately 6283.185&nbsp;mrad). Almost all [[Telescopic sight|scope sights]] for [[firearm]]s are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which are ''approximately'' equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "[[NATO]] mil" is defined as {{sfrac|6400}} of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1&nbsp;km away ({{sfrac|2{{pi}}|6400}} = 0.0009817... ≈ {{sfrac|1000}}).
|-
|[[Binary angular measurement|binary degree]] ||256||1°33′45″  || The ''binary degree'', also known as the ''[[binary radian]]'' or ''brad'' or ''binary angular measurement (BAM)''.<ref name="ooPIC"/> The binary degree is used in computing so that an angle can be efficiently represented in a single [[byte]] (albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2<sup>''n''</sup> equal parts for other values of ''n''.
<ref name="Hargreaves_2010"/> It is {{sfrac|256}} of a turn.<ref name="ooPIC"/>
|-
|{{anchor|Multiples of π}}{{pi}} radian||2||180° || The ''multiples of {{pi}} radians'' (MUL{{pi}}) unit is implemented in the [[Reverse Polish Notation|RPN]] scientific calculator [[WP&nbsp;43S]].<ref name="Bonin_2016"/> See also: [[IEEE 754 recommended operations]]
|-
|[[circular sector|quadrant]]||4||90°||One ''quadrant'' is a {{sfrac|4}}&nbsp;turn and also known as a ''[[right angle]]''. The quadrant is the unit in [[Euclid's Elements]]. In German, the symbol <sup>∟</sup> has been used to denote a quadrant. 1 quad = 90° = {{sfrac|{{pi}}|2}}&nbsp;rad = {{sfrac|4}} turn = 100&nbsp;grad.
|-
|[[circular sector|sextant]]||6||60°||The ''sextant'' was the unit used by the [[Babylonians]],<ref name="Jeans_1947"/><ref name="Murnaghan_1946"/> The degree, minute of arc and second of arc are [[sexagesimal]] subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the ''angle of the [[equilateral triangle]]'' or is {{sfrac|6}}&nbsp;turn. 1 Babylonian unit = 60° = {{pi}}/3&nbsp;rad ≈ 1.047197551&nbsp;rad.
|-
| hexacontade||60 ||6°||The ''hexacontade'' is a unit used by [[Eratosthenes]]. It equals 6°, so a whole turn was divided into 60 hexacontades.
|-
| [[pechus]]|| 144 to 180 || 2° to 2°30′ || The ''pechus'' was a [[Babylonian mathematics|Babylonian]] unit equal to about 2° or {{sfrac|2|1|2}}°.
|-
| diameter part || ≈376.991 || ≈0.95493° || The ''diameter part'' (occasionally used in Islamic mathematics) is {{sfrac|60}} radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
|-
| zam || 224 || ≈1.607° || In old Arabia, a [[Turn (geometry)|turn]] was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a [[Turn (geometry)|turn]] is 224 zam.
|}