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Unit systemSI derived unit
Unit ofAngle
Conversions
1 rad in ...... is equal to ...
turns   1/2π turn
degrees   180°/π ≈ 57.296°
An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.

The radian, denoted by the symbol rad, is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995) and the radian is now an SI derived unit.[1] The radian is defined in the SI as being a dimensionless unit with 1 rad = 1.[2] Its symbol is accordingly often omitted, especially in mathematical writing.

## Definition

One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle.[3] More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius. A right angle is exactly π/2 radians.[4]

A complete revolution is 2π radians (shown here with a circle of radius one and thus circumference 2π).

The magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees ≈ 57.295779513082320876 degrees.[5]

The relation 2π rad = 360° can be derived using the formula for arc length, ${\displaystyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}$, and by using a circle of radius 1. Since radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, ${\displaystyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}$. This can be further simplified to ${\displaystyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}$. Multiplying both sides by 360° gives 360° = 2π rad.

## Unit symbol

The International Bureau of Weights and Measures[4] and International Organization for Standardization[6] specify rad as the symbol for the radian. Alternative symbols used 100 years ago are c (the superscript letter c, for "circular measure"), the letter r, or a superscript R,[7] but these variants are infrequently used, as they may be mistaken for a degree symbol (°) or a radius (r). Hence a value of 1.2 radians would most commonly be written as 1.2 rad; other notations include 1.2 r, 1.2rad, 1.2c, or 1.2R.

In mathematical writing, the symbol "rad" is often omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used.

## Conversions

A chart to convert between degrees and radians
Conversion of common angles
1/24 turn π/12 rad 15° 16+2/3g
1/16 turn π/8 rad 22.5° 25g
1/12 turn π/6 rad 30° 33+1/3g
1/10 turn π/5 rad 36° 40g
1/8 turn π/4 rad 45° 50g
1/2π turn 1 rad c. 57.3° c. 63.7g
1/6 turn π/3 rad 60° 66+2/3g
1/5 turn 2π/5 rad 72° 80g
1/4 turn π/2 rad 90° 100g
1/3 turn 2π/3 rad 120° 133+1/3g
2/5 turn 4π/5 rad 144° 160g
1/2 turn π rad 180° 200g
3/4 turn 3π/2 rad 270° 300g
1 turn 2π rad 360° 400g

### Conversion between radians and degrees

As stated, one radian is equal to ${\displaystyle {180^{\circ }}/{\pi }}$. Thus, to convert from radians to degrees, multiply by ${\displaystyle {180^{\circ }}/{\pi }}$.

${\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}$

For example:

${\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}$
${\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}$
${\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}$

Conversely, to convert from degrees to radians, multiply by ${\displaystyle {\pi }/{180^{\circ }}}$.

${\displaystyle {\text{angle in radians}}={\text{angle in degrees}}\cdot {\frac {\pi }{180^{\circ }}}}$

For example:

${\displaystyle 1^{\circ }=1^{\circ }\cdot {\frac {\pi }{180^{\circ }}}\approx 0.0175{\text{ rad}}}$

${\displaystyle 23^{\circ }=23^{\circ }\cdot {\frac {\pi }{180^{\circ }}}\approx 0.4014{\text{ rad}}}$

Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2π.

#### Radian to degree conversion derivation

The length of circumference of a circle is given by ${\displaystyle 2\pi r}$, where ${\displaystyle r}$ is the radius of the circle.

So the following equivalent relation is true:

${\displaystyle 360^{\circ }r\iff 2\pi r}$ [Since a ${\displaystyle 360^{\circ }}$ sweep is needed to draw a full circle]

By the definition of radian, a full circle represents:

${\displaystyle {\frac {2\pi r}{r}}{\text{ rad}}}$
${\displaystyle =2\pi {\text{ rad}}}$

Combining both the above relations:

${\displaystyle 2\pi {\text{ rad}}=360^{\circ }}$
${\displaystyle \Rrightarrow 1{\text{ rad}}={\frac {360^{\circ }}{2\pi }}}$
${\displaystyle \Rrightarrow 1{\text{ rad}}={\frac {180^{\circ }}{\pi }}}$

${\displaystyle 2\pi }$ radians equals one turn, which is by definition 400 gradians (400 gons or 400g). So, to convert from radians to gradians multiply by ${\displaystyle 200^{\text{g}}/\pi }$, and to convert from gradians to radians multiply by ${\displaystyle \pi /200^{\text{g}}}$. For example,

${\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}$
${\displaystyle 50^{\text{g}}=50^{\text{g}}\cdot {\frac {\pi }{200^{\text{g}}}}\approx 0.7854{\text{ rad}}}$

Some common angles, measured in radians. All the large polygons in this diagram are regular polygons.

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 involving trigonometric functions can be elegantly stated, when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit formula

${\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}$

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

${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$[5]
${\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}$

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 ${\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}$, the evaluation of the integral ${\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}$ 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, when x is in radians, the Taylor series for sin x becomes:

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}$

If x were expressed in degrees, then the series would contain messy factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx / 180, so

${\displaystyle \sin x_{\mathrm {deg} }=\sin y_{\mathrm {rad} }={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}$

In a similar spirit, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) can be elegantly stated, when the functions' arguments are in radians (and messy otherwise).

## History

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714.[8][9] He described the radian in everything but name, and 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.[10]

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 1/60 radian. They also used sexagesimal subunits of the diameter part.[11]

The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. He had used the term as early as 1871, while in 1869, 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.[12][13][14] 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.[15]

### History as an SI unit

In 1960, the CGPM established the SI and the radian was classified as a "supplementary unit" along with the steradian. This special class was officially regarded "either as base units or as derived units".[16] In 1980 the Consultative Committee for Units (CCU) considered a proposal for making radians an SI base unit, using a constant α0 = 1 rad,[17][18] but turned it down to avoid an upheaval to current practice.[18] In October 1980 the CGPM decided that supplementary units were dimensionless derived units for which the CGPM allowed the freedom of using them or not using them in expressions for SI derived units,[16] on the basis that "[no formalism] exists which is at the same time coherent and convenient and in which the quantities plane angle and solid angle might be considered as base quantities" and that "[the possibility of treating the radian and steradian as SI base units] compromises the internal coherence of the SI based on only seven base units".[19] In 1995 the CGPM eliminated the class of supplementary units and defined the radian and the steradian as "dimensionless derived units, the names and symbols of which may, but need not, be used in expressions for other SI derived units, as is convenient".[20]

At the 2013 meeting of the CCU, Peter Mohr gave a presentation on alleged inconsistencies arising from defining the radian as a dimensionless unit rather than a base unit. CCU President Ian M. Mills declared this to be a "formidable problem" and the CCU Working Group on Angles and Dimensionless Quantities in the SI was established.[21] The CCU met most recently in 2021, but did not reach a consensus. A small number of members argued strongly that the radian should be a base unit, but the majority felt the status quo was acceptable or that the change would cause more problems than it would solve. A task group was established to "review the historical use of SI supplementary units and consider whether reintroduction would be of benefit", among other activities.[22][23]

## Dimensional analysis

The radian is defined as θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius. One radian corresponds to the angle for which s=r, hence 1 radian = 1 m/m.[24] However, ${\displaystyle \mathrm {rad} }$ is only to be used to express angles, not to express ratios of lengths in general.[4] A similar calculation using the area of a circular sector θ = 2A/r2 gives 1 radian as 1 m2/m2.[25] The key fact is that the radian is a dimensionless unit equal to 1. In SI 2019, the radian is defined accordingly as 1 rad = 1.[26] It is a long-established practice in mathematics and across all areas of science to make use of ${\displaystyle \mathrm {rad} =1}$.[27][28] In 1993 the AAPT Metric Committee specified that the radian should explicitly appear only when different numerical values for a quantity would be obtained when other angle measures were used, such as for angle measure (rad), angular speed (rad/s), angular acceleration (rad/s2), and torsional stiffness (N⋅m/rad), and not for other quantities, such as torque (N⋅m) and angular momentum (kg⋅m2/s).[29]

Giacomo Prando says "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations."[30] For example, a mass hanging by a string from a pulley will rise or drop by y= centimeters, where r is the radius of the pulley in centimeters and θ is the angle the pulley turns in radians. When multiplying r by θ the unit of radians of disappears from the result. Similarly in the formula for the angular velocity of a rolling wheel, ω=v/r, radians appear in the units of ω but not on the right hand side.[31] Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".[32] Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".[33]

At least a dozen scientists have made proposals to treat the radian as a base unit of measure defining its own dimension of "angle", as early as 1936 and as recently as 2022.[34][35][36] Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the area of a circle, πr2. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations".[37]

In particular Quincey identifies Torrens' proposal, to introduces a constant η equal to 1 inverse radian (1 rad−1) in a fashion similar to the introduction of the constant ε0.[37] With this change the formula for the angle subtended at the center of a circle, s = , is modified to become s = ηrθ, and the Taylor series for the sine of an angle θ becomes:[36][38]

${\displaystyle \operatorname {Sin} \theta =\sin(\eta \theta )=\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots .}$

The capitalized function ${\displaystyle \operatorname {Sin} }$ takes an argument with a dimension of angle, while ${\displaystyle \sin }$ retains its usual interpretation as a function on pure numbers.[38] This distinction can be ignored if the dimensions are clear from context.[36]

SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly 1 rad = 1. This radian convention allows the omission of η in mathematical formulas.[18]

A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.[36] Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.[39] For example, the Boost units library defines angle units with a plane_angle dimension.[40]

## Use in physics

The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically measured in radians per second (rad/s). One revolution per second is equal to 2π radians per second.

Similarly, angular acceleration is often measured in radians per second per second (rad/s2).

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

Likewise, the phase difference of two waves can also be measured in radians. For example, if the phase difference of two waves is (n⋅2π) radians, where n is an integer, they are considered in phase, whilst if the phase difference of two waves is (n⋅2π + π), where n is an integer, they are considered in antiphase.

## SI multiples

There are 2π × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under 1/6283 of the angle subtended by a full circle. This "real" unit of angular measurement of a circle is in use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.

An approximation of the milliradian (0.001 rad) is used by NATO and other military organizations in gunnery and targeting. Each angular mil represents 1/6400 of a circle and is 15/8% or 1.875% smaller than the 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 1/2000π; for example Sweden used the 1/6300 streck and the USSR used 1/6000. Being based on the milliradian, the NATO mil subtends roughly 1 m at a range of 1000 m (at such small angles, the curvature is negligible).

Prefixes smaller than milli- are useful in measuring extremely small angles. 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 the arc second, which is π/648,000 rad (around 4.8481 microradians).

## References

1. ^ "Resolution 8 of the CGPM at its 20th Meeting (1995)". Bureau International des Poids et Mesures. Archived from the original on 2018-12-25. Retrieved 2014-09-23.
2. ^ International Bureau of Weights and Measures 2019, p. 151: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, as dimensionless derived units."
3. ^ Protter, Murray H.; Morrey, Charles B., Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, p. APP-4, LCCN 76087042
4. ^ a b c
5. ^ a b Weisstein, Eric W. "Radian". mathworld.wolfram.com. Retrieved 2020-08-31.
6. ^
7. ^ Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter VII. The General Angle [55] Signs and Limitations in Value. Exercise XV.". Written at Ann Arbor, Michigan, USA. Trigonometry. Vol. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 73. Retrieved 2017-08-12.
8. ^ O'Connor, J. J.; Robertson, E. F. (February 2005). "Biography of Roger Cotes". The MacTutor History of Mathematics. Archived from the original on 2012-10-19. Retrieved 2006-04-21.
9. ^ 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, top of page 95. From page 95: After stating that 180° corresponds to a length of π (3.14159…) along a unit circle (i.e., π radians), Smith writes: "Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. " (Whence the unit of trigonometric measure, 57.2957795130… [degrees per radian], will appear.)
10. ^ Isaac Todhunter, Plane Trigonometry: For the Use of Colleges and Schools, p. 10, Cambridge and London: MacMillan, 1864 OCLC 500022958
11. ^ Luckey, Paul (1953) [Translation of 1424 book]. Siggel, A. (ed.). Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi [Treatise on the Circumference of al-Kashi]. Berlin: Akademie Verlag. p. 40.
12. ^ Cajori, Florian (1929). History of Mathematical Notations. Vol. 2. Dover Publications. pp. 147–148. ISBN 0-486-67766-4.
13. ^
14. ^ Miller, Jeff (Nov 23, 2009). "Earliest Known Uses of Some of the Words of Mathematics". Retrieved Sep 30, 2011.
15. ^ Frederick Sparks, Longmans' School Trigonometry, p. 6, London: Longmans, Green, and Co., 1890 OCLC 877238863 (1891 edition)
16. ^ a b Nelson, Robert A. (March 1984). "The supplementary units". The Physics Teacher. 22 (3): 188–193. doi:10.1119/1.2341516.
17. ^ Report of the 7th meeting (PDF) (in French), Consultative Committee for Units, 1980, pp. 6–7
18. ^ a b c
19. ^ International Bureau of Weights and Measures 2019, pp. 174–175.
20. ^
21. ^ Consultative Committee for Units (11–12 June 2013). Report of the 21st meeting to the International Committee for Weights and Measures (Report). pp. 18–20.
22. ^ Consultative Committee for Units (21–23 September 2021). Report of the 25th meeting to the International Committee for Weights and Measures (Report). pp. 16–17.
23. ^ "CCU Task Group on angle and dimensionless quantities in the SI Brochure (CCU-TG-ADQSIB)". BIPM. Retrieved 26 June 2022.
24. ^ International Bureau of Weights and Measures 2019, p. 151: "One radian corresponds to the angle for which s = r"
25. ^ Quincey 2016, p. 844: "Also, as alluded to in Mohr & Phillips 2015, the radian can be defined in terms of the area A of a sector (A = 1/2 θ r2), in which case it has the units m2⋅m−2."
26. ^ International Bureau of Weights and Measures 2019, p. 151: "One radian corresponds to the angle for which s = r, thus 1 rad = 1."
27. ^
28. ^ Bridgman, Percy Williams (1922). Dimensional analysis. New Haven : Yale University Press. Angular amplitude of swing [...] No dimensions.
29. ^ Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit". The Physics Teacher. 31 (2): 84–87. Bibcode:1993PhTea..31...84A. doi:10.1119/1.2343667.
30. ^ Prando, Giacomo (August 2020). "A spectral unit". Nature Physics. 16 (8): 888–888. doi:10.1038/s41567-020-0997-3.
31. ^ Leonard, William J. (1999). Minds-on Physics: Advanced topics in mechanics. Kendall Hunt. p. 262. ISBN 978-0-7872-5412-4.
32. ^ French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)". The Physics Teacher. 30 (5): 260–261. doi:10.1119/1.2343535.
33. ^ Oberhofer, E. S. (March 1992). "What happens to the 'radians'?". The Physics Teacher. 30 (3): 170–171. doi:10.1119/1.2343500.
34. ^
35. ^
36. ^ a b c d Quincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base quantity in the SI". Metrologia. 53 (3): 998–1002. arXiv:1604.02373. Bibcode:2016Metro..53..998Q. doi:10.1088/0026-1394/53/3/998. S2CID 119294905.
37. ^ a b
38. ^ a b
39. ^ Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems". Metrologia. 54 (4): 454–460. arXiv:1705.03765. Bibcode:2017Metro..54..454Q. doi:10.1088/1681-7575/aa7160. S2CID 119418270.
40. ^ Schabel, Matthias C.; Watanabe, Steven. "Boost.Units FAQ - 1.79.0". www.boost.org. Retrieved 5 May 2022. Angles are treated as units
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