Editing Angle (section)

==Measuring angles{{anchor|Measurement}}==<!-- linked from [[Degree (angle)]] -->
{{see also|Angle measuring instrument}}
[[File:Basic angle in circle.svg|thumb|300x300px|The angle size <math>\theta</math> can be measured as s/r radians or s/C turns]]
Measurement of angles is intrinsically linked with circles and rotation. An angle is measured by placing it within a circle of any size, with the vertex at the circle's centre and the sides intersecting the perimeter. 

An [[circular arc|arc]] <var>s</var> is formed as the shortest distance on the perimeter between the two points of intersection, which is said to be the arc [[Subtended angle|subtended]] by the angle. 

The length of ''s'' can be used to measure the angle's size <math>\theta</math>, however as ''s'' is dependent on the size of the circle chosen, it must be adjusted so that any arbitrary circle will give the same measure of angle. This can be done in two ways: by taking the ratio to either the radius ''r'' or circumference ''C'' of the circle.

The ratio of the length <var>s</var> by the radius <var>r</var> is the number of [[radian]]s in the angle, while the ratio of length <var>s</var> by the circumference <var>C</var> is the number of [[Turn (angle)|turns]]:<ref name="SIBrochure9thEd">{{citation |author=International Bureau of Weights and Measures |title=The International System of Units (SI) |date=20 May 2019 |url=https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf |archive-url=https://web.archive.org/web/20211018184555/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9.pdf/fcf090b2-04e6-88cc-1149-c3e029ad8232 |archive-date=18 October 2021 |url-status=live |edition=9th |isbn=978-92-822-2272-0 |author-link=New SI}}</ref>
<math display="block"> \theta = \frac{s}{r} \, \mathrm{rad}. </math><math display="block"> \theta = \frac{s}{ C} \ = \frac{s}{2\pi r} \, \mathrm{turns} </math>

[[File:Angle measure.svg|right|thumb|The measure of angle {{math|''θ''}} is {{nowrap|{{sfrac|''s''|''r''}} radians}}.]]

The value of {{math|''θ''}} thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios <math>\frac{s}{r}</math>and <math>\frac{s}{C}</math> are unaltered.{{refn|group="nb"|This approach requires, however, an additional proof that the measure of the angle does not change with changing radius {{math|''r''}}, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.<ref name="Dimitric_2012"/>}}

Angles of the same size are said to be ''equal'' ''congruent'' or ''equal in measure''.

=== Units ===
In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including the [[degree (angle)|degree]] ( ° ) 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> 

Conversion between units may be obtained by multiplying the anglular measure in one unit by a conversion constant of the form <math>\frac{k_a}{k_b}</math> where <math>{k_a}</math> and <math>{k_b}</math> are the measures of a complete turn expressed in units a and b. For example, {{nowrap|1= ''k'' = 360°}} for [[degree (angle)|degrees]] or 400 grad for [[gradian]]s):<math display="block"> \theta_\deg = \frac{360}{2\pi} \cdot \theta </math>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.
|}

===Dimensional analysis===
{{Further|Radian#Dimensional analysis}}
In mathematics and 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]]. For example, when one measures an angle in radians by dividing the arc length by the radius, one is essentially dividing a length by another length, and the units of length cancel each other out. Therefore the result—the angle—doesn't have a physical "dimension" like meters or seconds. This holds true with all angle units, such as radians, degrees, or turns—they all represent a pure number quantifying how much something has turned. This is why, in many equations, angle units seem to "disappear" during calculations, which can sometimes be a bit confusing.

This disappearing act, while mathematically convenient, has led to significant discussion among scientists and teachers, as it can be tricky to explain and feels inconsistent. To address this, some scientists have suggested treating the angle as having its own fundamental dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, making the dimensional analysis more straightforward. However, this approach would also require changing many well-known mathematical and physics formulas, making them longer and perhaps a bit less familiar. For now, the established practice is to consider angles dimensionless, understanding that while units like radians are important for expressing the angle's magnitude, they don't carry a physical dimension in the same way that meters or kilograms do.

===Signed angles ===
{{main|Angle of rotation}}
{{see also|Sign (mathematics)#Angles|Euclidean space#Angle}}
[[File:Angles on the unit circle.svg|right|thumb|Measuring from the [[x-axis]], angles on the [[unit circle]] count as positive in the [[counterclockwise]] direction, and negative in the [[clockwise]] direction.]]

An angle denoted as {{math|∠BAC}} might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, It is therefore frequently helpful to impose a convention that allows positive and negative angular values to represent [[Orientation (geometry)|orientations]] and/or [[Rotation (mathematics)|rotations]] in opposite directions or "sense" relative to some reference.

In a two-dimensional [[Cartesian coordinate system]], an angle is typically defined by its two sides, with its vertex at the origin. The ''initial side'' is on the positive [[x-axis]], while the other side or ''terminal side'' is defined by the measure from the initial side in radians, degrees, or turns, with ''positive angles'' representing rotations toward the positive [[y-axis]] and ''negative angles'' representing rotations toward the negative ''y''-axis. When Cartesian coordinates are represented by ''standard position'', defined by the ''x''-axis rightward and the ''y''-axis upward, positive rotations are [[anticlockwise]], and negative cycles are [[clockwise]].

In many contexts, an angle of −''θ'' is effectively equivalent to an angle of "one full turn minus ''θ''". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360°&nbsp;−&nbsp;45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an [[Orientability|orientation]], which is typically determined by a [[Normal (geometry)|normal vector]] passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In [[navigation]], [[bearing (navigation)|bearings]] or [[azimuth]] are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

===Equivalent angles===
* Angles that have the same measure (i.e., the same magnitude) are said to be ''equal'' or ''[[Congruence (geometry)|congruent]]''. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all ''right angles'' are equal in measure).
* Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called ''coterminal angles''.
* The ''reference angle'' (sometimes called ''related angle'') for any angle ''θ'' in standard position is the positive acute angle between the terminal side of ''θ'' and the x-axis (positive or negative).<ref>{{cite web|url=http://www.mathwords.com/r/reference_angle.htm|title=Mathwords: Reference Angle|website=www.mathwords.com|access-date=26 April 2018|url-status=live|archive-url=https://web.archive.org/web/20171023035017/http://www.mathwords.com/r/reference_angle.htm|archive-date=23 October 2017}}</ref><ref>{{cite book |last1=McKeague |first1=Charles P. |title=Trigonometry |date=2008 |publisher=Thomson Brooks/Cole |location=Belmont, CA |isbn=978-0495382607 |page=110 |edition=6th}}</ref> Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude [[modulo]] {{sfrac|2}} turn, 180°, or {{math|π}} radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

===Related quantities===
For an angular unit, it is definitional that the [[angle addition postulate]] holds. Some quantities related to angles where the angle addition postulate does not hold include:
* The ''[[slope]]'' or ''gradient'' is equal to the [[tangent (trigonometric function)|tangent]] of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.
* The ''[[spread (rational trigonometry)|spread]]'' between two lines is defined in [[rational geometry]] as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
* Although done rarely, one can report the direct results of [[trigonometric functions]], such as the [[sine]] of the angle.