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{{about||a measure of ionizing radiation|Rad (unit)|other uses|Radian (disambiguation)}}
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{{Infobox Unit
| name = Radian  
| standard = [[SI derived unit]]
| quantity = [[Angle]]
| symbol = rad
| symbol2 = {{sup|c}}
| symbol3 = r
| symbol4 = {{sup|R}}<ref name="Hall_1909"/>
| extralabel   = In units
| extradata    = [[Dimensionless quantity|Dimensionless]] with an arc length equal to the radius, i.e. 1&nbsp;{{sfrac|m|m}}
| units1     = [[milliradian]]s
| inunits1   = 1,000 milliradians
| units2     = [[Turn (unit)|turn]]s
| inunits2   = {{sfrac|1|2{{pi}}}} turn
| units3     = [[Degree (angle)|degree]]s
| inunits3   = {{sfrac|180|{{pi}}}} ≈ 57.296°
| units4     = [[Gon (unit)|gon]]s
| inunits4   = {{sfrac|200|{{pi}}}} ≈ 63.662<sup>g</sup>
}}
[[File:Circle radians.gif|thumb|right|300px|An arc of a [[circle]] with the same length as the [[radius]] of that circle subtends an ''angle'' of 1&nbsp;radian. The circumference subtends an angle of 2[[pi|{{math|&pi;}}]] radians.]]
The '''radian''' is the [[International System of Units|SI unit]] for measuring [[angle]]s, and is the standard unit of angular measure used in many areas of [[mathematics]].  The length of an arc of a [[unit circle]] is numerically equal to the measurement in radians of the [[angle]] that it [[wikt:subtend|subtend]]s; one radian is just under 57.3 [[Degree (angle)|degrees]] (expansion at {{OEIS2C|A072097}}). The unit was formerly an [[SI supplementary unit]], but this category was abolished in 1995 and the radian is now considered an [[SI derived unit]].<ref>{{cite web |title=Resolution 8 of the CGPM at its 20th Meeting (1995) |url=http://www.bipm.org/en/CGPM/db/20/8/ |access-date=2014-09-23 |publisher=[[Bureau International des Poids et Mesures]]}}</ref>

The symbol '''rad''' can be used to represent the radian (for example, an angle of {{math|1.3}} radians would be written as {{math|1.3 rad}}), but this is often omitted, especially in mathematical writing.

== Definition ==

Radian describes the plane [[angle]] [[subtended]] by a circular [[Arc (geometry)|arc]] as the length of the arc divided by the [[radius]] of the arc. One radian is the angle subtended at the center of a [[circle]] by an [[Arc (geometry)|arc]] that is equal in length to the [[radius]] of the circle. More generally, the [[magnitude (mathematics)|magnitude]] in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, {{nowrap|1=''[[θ]]'' = ''s'' / ''r''}}, where ''θ'' is the subtended angle in radians, ''s'' is arc length, and ''r'' is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, {{nowrap|1=''s'' = ''rθ''}}.

As the ratio of two lengths, the radian is a "[[pure number]]" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted{{fact|date=September 2019}}. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol [[°]] is used. The radian is defined as 1.<ref>ISO 80000-3:2006</ref>

[[File:2pi-unrolled.gif|thumb|right|A complete revolution is 2{{pi}} radians (shown here with a circle of radius one and thus [[circumference]] 2{{pi}}).]]
It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or {{nowrap|2[[pi|{{pi}}]]''r'' / ''r''}}, or 2{{pi}}. Thus 2{{pi}} radians is equal to 360 degrees, meaning that one radian is equal to 180/{{pi}} degrees.

The relation <math>2\pi\text{ rad}=360^{\circ}</math> can be derived using the formula for [[arc length]]. Taking the formula for arc length, or <math>\ell_{arc}=2\pi r\left(\frac{\theta}{360^{\circ}}\right)</math>. Assuming a unit circle; the radius is therefore one. Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that <math>1=2\pi\left(\frac{1\text{ rad}}{360^{\circ}}\right)</math>. This can be further simplified to <math>1=\frac{2\pi\text{ rad}}{360^{\circ}}</math>. Multiplying both sides by <math>360^{\circ}</math> gives <math>360^{\circ}=2\pi\text{ rad}</math>.

== History ==

The concept of radian measure, as opposed to the degree of an angle, is normally credited to [[Roger Cotes]] in 1714.<ref>{{cite web|url = http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html | title = Biography of Roger Cotes | work=The MacTutor History of Mathematics |date = February 2005 | last1= O'Connor|first1= J. J. |first2= E. F. |last2=Robertson}}</ref><ref>Roger Cotes died in 1716.  By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, ''Harmonia mensurarum'' … .  In a chapter of editorial comments by Smith, he gives, for the first time, the value of one radian in degrees.  See:  Roger Cotes with Robert Smith, ed., ''Harmonia mensurarum'' … (Cambridge, England:  1722), chapter: Editoris notæ ad Harmoniam mensurarum, [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=RA1-PA95#v=onepage&q&f=false top of page 95].  From page 95:  After stating that 180° corresponds to a length of {{pi}} (3.14159…) along a unit circle (i.e., {{pi}} radians), Smith writes:  ''"Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. "'' (Whence the unit of trigonometric measure, 57.2957795130… [degrees per radian], will appear.)</ref> He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure. Prior to the term ''radian'' becoming widespread, the unit was commonly called ''circular measure'' of an angle.<ref>Isaac Todhunter, ''Plane Trigonometry: For the Use of Colleges and Schools'', [https://books.google.co.uk/books?id=bo5FAAAAcAAJ&pg=PA10 p. 10], Cambridge and London: MacMillan, 1864 {{OCLC|500022958}}</ref>

The idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, [[al-Kashi]] (c. 1400) used so-called ''diameter parts'' as units where one diameter part was {{sfrac|1|60}} radian and they also used sexagesimal subunits of the diameter part.<ref>{{cite book|first=Paul|last= Luckey|editor-first=A. |editor-last=Siggel|location=Berlin|publisher= Akademie Verlag| origyear=Translation of 1424 book|year=1953| title=Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi|trans-title=Treatise on the Circumference of al-Kashi| number=6|pages= 40}}</ref>

The term ''radian'' first appeared in print on 5 June 1873, in examination questions set by [[James Thomson (engineer)|James Thomson]] (brother of [[Lord Kelvin]]) at [[Queen's University Belfast|Queen's College]], [[Belfast]]. He had used the term as early as 1871, while in 1869, [[Thomas Muir (mathematician)|Thomas Muir]], then of the [[University of St Andrews]], vacillated between the terms ''rad'', ''radial'', and ''radian''. In 1874, after a consultation with James Thomson, Muir adopted ''radian''.<ref>{{cite book| author-link=Florian Cajori| first=Florian| last=Cajori| year=1929| title=History of Mathematical Notations| volume=2| pages=147–148| isbn=0-486-67766-4| url-access=registration| url=https://archive.org/details/historyofmathema00cajo_0}}</ref><ref>{{cite journal| journal=Nature| year=1910| volume= 83| pages=156|doi=10.1038/083156a0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2110|bibcode = 1910Natur..83..156M | url=https://zenodo.org/record/1429528/files/article.pdf}}{{cite journal| journal=Nature| year=1910| volume= 83| pages=217|doi=10.1038/083217c0| title=The Term "Radian" in Trigonometry| last1=Thomson| first1=James| issue=2112|bibcode = 1910Natur..83..217T | url=https://zenodo.org/record/1429530}}{{cite journal| journal=Nature| year=1910| volume= 83| pages=459–460|doi=10.1038/083459d0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2120|bibcode = 1910Natur..83..459M }}</ref><ref>{{cite web|url=http://jeff560.tripod.com/r.html|date=Nov 23, 2009| accessdate=Sep 30, 2011|last=Miller|first=Jeff |title= Earliest Known Uses of Some of the Words of Mathematics}}</ref> The name ''radian'' was not universally adopted for some time after this.  ''Longmans' School Trigonometry'' still called the radian ''circular measure'' when published in 1890.<ref>Frederick Sparks, ''Longmans' School Trigonometry'', p. 6, London: Longmans, Green, and Co., 1890 {{oclc|877238863}} (1891 edition)</ref>
== Unit symbol ==
The [[International Bureau of Weights and Measures]]<ref>[https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf 2019 BIPM Brochure]</ref> and [[International Organization for Standardization]]<ref>[https://www.iso.org/standard/31888.html ISO 80000-3:2006 Quantities and Units - Space and Time]</ref> specify '''rad''' as the symbol for the radian. Alternative symbols used 100 years ago are <sup>c</sup> (the superscript letter c, for "circular measure"), the letter r, or a superscript {{sup|R}},<ref name="Hall_1909"/> but these variants are infrequently used as they may be mistaken for a [[degree symbol]] (°) or a radius (r). So, for example, a value of 1.2 radians would most commonly be written as 1.2&nbsp;rad; other notations include 1.2&nbsp;r, 1.2{{sup|rad}}, 1.2{{sup|c}}, or 1.2{{sup|R}}.

== Conversions ==
[[File:Degree-Radian Conversion.svg|thumb|300px|A chart to convert between degrees and radians]]

{{Table of angles}}

===Conversion between radians and degrees===
As stated, one radian is equal to 180/{{pi}} degrees. Thus, to convert from radians to degrees, multiply by 180/{{pi}}.

:<math> \text{angle in degrees} = \text{angle in radians} \cdot \frac {180^\circ} {\pi}</math>

For example:
:<math>1 \text{ rad} = 1 \cdot \frac {180^\circ} {\pi} \approx 57.2958^\circ </math>
:<math>2.5 \text{ rad} = 2.5 \cdot \frac {180^\circ} {\pi} \approx 143.2394^\circ </math>
:<math>\frac {\pi} {3} \text{ rad} = \frac {\pi} {3} \cdot \frac {180^\circ} {\pi} = 60^\circ </math>

Conversely, to convert from degrees to radians, multiply by {{pi}}/180.

:<math> \text{angle in radians} = \text{angle in degrees} \cdot \frac {\pi} {180^\circ}</math>

For example:

:<math>1^\circ = 1 \cdot \frac {\pi} {180^\circ} \approx 0.0175 \text{ rad}</math>
<math>23^\circ = 23 \cdot \frac {\pi} {180^\circ} \approx 0.4014 \text{ rad}</math>

Radians can be converted to [[Turn (geometry)|turns]] (complete revolutions) by dividing the number of radians by 2{{pi}}.

====Radian to degree conversion derivation====

The length of circumference of a circle is given by <math>2\pi r</math>, where <math>r</math> is the radius of the circle.

So the following equivalent relation is true:

<math>360^\circ \iff 2\pi r</math>{{pad|4em}}[Since a <math>360^\circ</math> sweep is needed to draw a full circle]

By the definition of radian, a full circle represents:

:<math>\frac{2\pi r}{r} \text{ rad}</math>

:<math>= 2\pi \text{ rad}</math>

Combining both the above relations:

:<math>2\pi \text{ rad} = 360^\circ</math>

:<math>\Rrightarrow 1 \text{ rad} = \frac{360^\circ}{2\pi}</math>

:<math>\Rrightarrow 1 \text{ rad} = \frac{180^\circ}{\pi}</math>

===Conversion between radians and gradians===

<math>2\pi</math> radians equals one [[turn (geometry)|turn]], which is by definition 400 [[Grad (angle)|gradians]] (400 [[gon (angle)|gon]]s or 400<sup>g</sup>). So, to convert from radians to gradians multiply by <math>200/\pi</math>, and to convert from gradians to radians multiply by <math>\pi/200</math>. For example,

:<math>1.2 \text{ rad} = 1.2 \cdot \frac {200^\text{g}} {\pi} \approx 76.3944^\text{g}</math>
:<math>50^\text{g} = 50 \cdot \frac {\pi} {200^\text{g}} \approx 0.7854 \text{ rad}</math>

==Advantages of measuring in radians==
[[File:Radian-common.svg|thumb|357px|right|Some common angles, measured in radians. All the large polygons in this diagram are [[regular polygon]]s.]]

In [[calculus]] and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results.

Most notably, results in [[analysis (mathematics)|analysis]] involving [[trigonometric function]]s are simple and elegant when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple [[limit of a function|limit]] formula

:<math>\lim_{h\rightarrow 0}\frac{\sin h}{h}=1,</math>

which is the basis of many other identities in mathematics, including

:<math>\frac{d}{dx} \sin x = \cos x</math>
:<math>\frac{d^2}{dx^2} \sin x = -\sin x.</math>

Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the differential equation <math> \frac{d^2 y}{dx^2} = -y </math>, the evaluation of the integral <math> \int \frac{dx}{1+x^2} </math>, and so on). In all such cases it is found that the arguments to the functions are most naturally written in the form that corresponds, in geometrical contexts, to the radian measurement of angles.

The trigonometric functions also have simple and elegant series expansions when radians are used; for example, the following [[Taylor series]] for sin&nbsp;''x'' :

:<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots .</math>

If ''x'' were expressed in degrees then the series would contain messy factors involving powers of {{pi}}/180: if ''x'' is the number of degrees, the number of radians is {{nowrap|1=''y'' = {{pi}}''x'' / 180}}, so

:<math>\sin x_\mathrm{deg} = \sin y_\mathrm{rad} = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3\ \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5\ \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7\ \frac{x^7}{7!} + \cdots .</math>

Mathematically important relationships between the sine and cosine functions and the [[exponential function]] (see, for example, [[Euler's formula]]) are, again, elegant when the functions' arguments are in radians and messy otherwise.

== Dimensional analysis ==

Although the radian is a unit of measure, it is a [[dimensionless quantity]]. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.

Although [[Polar coordinates|polar]] and [[spherical coordinates]] use radians to describe coordinates in two and three dimensions, the unit is derived from the radius coordinate, so the angle measure is still dimensionless.<ref>For a debate on this meaning and use see:
{{cite journal|doi=10.1119/1.18616|title=Angles—Let's treat them squarely|year=1997|last1=Brownstein|first1=K. R.|journal=American Journal of Physics|volume=65|issue=7|pages=605–614|bibcode = 1997AmJPh..65..605B }},
{{cite journal|title=Angles as a fourth fundamental quantity|year=1962|last1=Romain|first1=J.E.|journal=Journal of Research of the National Bureau of Standards Section B|volume=66B|issue=3|pages=97|doi=10.6028/jres.066B.012}},
{{cite journal|doi=10.1119/1.18964|title=Dimensional angles and universal constants|year=1998|last1=LéVy-Leblond|first1=Jean-Marc|journal=American Journal of Physics|volume=66|issue=9|pages=814–815|bibcode = 1998AmJPh..66..814L }}, and {{cite journal|doi=10.1119/1.19185|title=Units—SI-Only, or Multicultural Diversity?|year=1999|last1=Romer|first1=Robert H.|journal=American Journal of Physics|volume=67|issue=1|pages=13–16 |bibcode = 1999AmJPh..67...13R }}</ref>

==Use in physics==
The radian is widely used in [[physics]] when angular measurements are required. For example, [[angular velocity]] is typically measured in [[Radian per second|radians per second]] (rad/s). One revolution per second is equal to 2{{pi}} radians per second.

Similarly, [[angular acceleration]] is often measured in radians per second per second (rad/s<sup>2</sup>).

For the purpose of dimensional analysis, the units of angular velocity and angular acceleration are s<sup>−1</sup> and s<sup>−2</sup> respectively.

Likewise, the [[phase difference]] of two waves can also be measured in radians. For example, if the phase difference of two waves is (''k''⋅2{{pi}}) radians, where ''k'' is an integer, they are considered in [[phase (waves)|phase]], whilst if the phase difference of two waves is ({{nowrap|''k''⋅2{{pi}} + {{pi}}}}), where ''k'' is an integer, they are considered in antiphase.

== SI multiples ==
[[Metric prefix]]es have limited use with radians, and none in mathematics. A [[milliradian]] (mrad) is a thousandth of a radian and a microradian (μrad) is a millionth of a radian, i.e. {{nowrap|1=1 rad = 10<sup>3</sup> mrad = 10<sup>6</sup> μrad}}.

There are 2[[pi|{{pi}}]] × 1000 milliradians (≈ 6283.185&nbsp;mrad) in a circle. So a trigonometric milliradian is just under {{sfrac|1|6283}} of a circle. This “real” trigonometric unit of angular measurement of a circle is in use by [[telescopic sight]] manufacturers using [[Stadiametric rangefinding|(stadiametric) rangefinding]] in [[reticle]]s. The [[beam divergence|divergence]] of [[laser]] beams is also usually measured in milliradians.

An approximation of the trigonometric milliradian (0.001&nbsp;rad) is used by [[NATO]] and other military organizations in [[gun]]nery and [[Sniper#Targeting|targeting]]. Each angular mil represents {{sfrac|1|6400}} of a circle and is {{sfrac|15|8}}% or 1.875% smaller than the trigonometric milliradian. For the small angles typically found in targeting work, the convenience of using the number 6400 in calculation outweighs the small mathematical errors it introduces. In the past, other gunnery systems have used different approximations to {{sfrac|1|2000{{pi}}}}; for example Sweden used the {{sfrac|1|6300}} ''streck'' and the USSR used {{sfrac|1|6000}}. Being based on the milliradian, the NATO mil subtends roughly 1&nbsp;m at a range of 1000&nbsp;m (at such small angles, the curvature is negligible).

Smaller units like microradians (μrad) and nanoradians (nrad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is [[arc second]], which is {{sfrac|{{pi}}|648,000}}&nbsp;rad (around 4.8481&nbsp;microradians). Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.

== See also ==
* [[Angular frequency]]
* [[Steradian]], a higher-dimensional analog of the radian which measures solid angle
* [[Trigonometry]]

== Notes and references ==
{{Notelist}}

{{Reflist|refs=
<ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |author-first1=Arthur Graham |author-last1=Hall |author-first2=Fred Goodrich |author-last2=Frink |date=January 1909 |location=Ann Arbor, Michigan, USA |chapter=Chapter VII. The General Angle [55] Signs and Limitations in Value. Exercise XV. |publisher=[[Henry Holt and Company]] / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA |publication-place=New York, USA |page=73 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n88/mode/1up |access-date=2017-08-12 |url-status=live}}</ref>
}}

==External links==
{{Wikibooks|Trigonometry/Radian and degree measures}}
{{Wiktionary|radian}}
*{{Commonscatinline}}
*[http://mathworld.wolfram.com/Radian.html Radian] at [[MathWorld]]

{{SI units}}

[[Category:Natural units]]
[[Category:SI derived units]]
[[Category:UCUM base units]]
[[Category:Pi]]
[[Category:Units of plane angle]]