Radian

For a measure of ionizing radiation, see Rad (unit). For other uses, see Radian (disambiguation).

Radian
Unit system	SI derived unit
Unit of	Angle
Symbol	rad, c or r
Conversions
1 rad in ...	... is equal to ...
milliradians	1000 mrad
turns	1/2π turn
degrees	180°/π ≈ 57.296°
gradians	200g/π ≈ 63.662g

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. The radian is defined in the SI as being a dimensionless unit with 1 rad = 1. 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. 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.

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.

The relation 2π rad = 360° can be derived using the formula for arc length, $\ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)$, 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, $1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)$. This can be further simplified to $1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}$. Multiplying both sides by 360° gives 360° = 2π rad.

Unit symbol

The International Bureau of Weights and Measures and International Organization for Standardization 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, 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, 1.2^rad, 1.2^c, or 1.2^R.

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

Turns	Radians	Degrees	Gradians, or gons
0 turn	0 rad	0°	0g
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 ${180^{\circ }}/{\pi }$. Thus, to convert from radians to degrees, multiply by ${180^{\circ }}/{\pi }$.

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

For example:

$1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }$

$2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }$

${\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }$

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

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

For example:

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

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

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 $2\pi r$, where $r$ is the radius of the circle.

So the following equivalent relation is true:

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

By the definition of radian, a full circle represents:

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

$=2\pi {\text{ rad}}$

Combining both the above relations:

$2\pi {\text{ rad}}=360^{\circ }$

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

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

Conversion between radians and gradians

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

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

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

Advantages of measuring in radians

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

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

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

${\frac {d}{dx}}\sin x=\cos x$

${\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.$

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 ${\tfrac {d^{2}y}{dx^{2}}}=-y$, the evaluation of the integral $\textstyle \int {\frac {dx}{1+x^{2}}},$ 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:

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

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

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

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. 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.

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.

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. 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.

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". In 1980 the Consultative Committee for Units (CCU) considered a proposal for making radians an SI base unit, using a constant α₀ = 1 rad, but turned it down to avoid an upheaval to current practice. 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, 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". 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".

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. 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.

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. However, $\mathrm {rad}$ is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using the area of a circular sector θ = 2A/r² gives 1 radian as 1 m²/m². 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. It is a long-established practice in mathematics and across all areas of science to make use of $\mathrm {rad} =1$. 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).

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. 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".

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. With this change the formula for the angle subtended at the center of a circle, s = rθ, is modified to become s = ηrθ, and the Taylor series for the sine of an angle θ becomes:

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

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

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.

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. Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal. For example, the Boost units library defines angle units with a plane_angle dimension.

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

Metric prefixes for submultiples are used with radians. A milliradian (mrad) is a thousandth of a radian and a microradian (μrad) is a millionth of a radian, i.e. 1 rad = 10³ mrad = 10⁶ μrad.

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).

See also

• Angular frequency
• Minute and second of arc
• Steradian, a higher-dimensional analog of the radian which measures solid angle
• Trigonometry

References

• International Bureau of Weights and Measures (20 May 2019), The International System of Units (SI) (PDF) (9th ed.), ISBN 978-92-822-2272-0, archived (PDF) from the original on 8 May 2021
• Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly". American Journal of Physics. 4 (4): 175–179. Bibcode:1936AmJPh...4..175B. doi:10.1119/1.1999110.
• Romain, Jacques E. (July 1962). "Angle as a fourth fundamental quantity". Journal of Research of the National Bureau of Standards, Section B: Mathematics and Mathematical Physics. 66B (3): 97. doi:10.6028/jres.066B.012.
• Eder, W E (January 1982). "A Viewpoint on the Quantity "Plane Angle"". Metrologia. 18 (1): 1–12. Bibcode:1982Metro..18....1E. doi:10.1088/0026-1394/18/1/002.
• Torrens, A B (1 January 1986). "On Angles and Angular Quantities". Metrologia. 22 (1): 1–7. Bibcode:1986Metro..22....1T. doi:10.1088/0026-1394/22/1/002.
• Brownstein, K. R. (July 1997). "Angles—Let's treat them squarely". American Journal of Physics. 65 (7): 605–614. Bibcode:1997AmJPh..65..605B. doi:10.1119/1.18616.
• Lévy-Leblond, Jean-Marc (September 1998). "Dimensional angles and universal constants". American Journal of Physics. 66 (9): 814–815. Bibcode:1998AmJPh..66..814L. doi:10.1119/1.18964.
• Foster, Marcus P (1 December 2010). "The next 50 years of the SI: a review of the opportunities for the e-Science age". Metrologia. 47 (6): R41–R51. doi:10.1088/0026-1394/47/6/R01.
• Mohr, Peter J; Phillips, William D (1 February 2015). "Dimensionless units in the SI". Metrologia. 52 (1): 40–47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40.
• Quincey, Paul (1 April 2016). "The range of options for handling plane angle and solid angle within a system of units". Metrologia. 53 (2): 840–845. Bibcode:2016Metro..53..840Q. doi:10.1088/0026-1394/53/2/840.
• Mills, Ian (1 June 2016). "On the units radian and cycle for the quantity plane angle". Metrologia. 53 (3): 991–997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991.
• Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem". Metrologia. 58 (5): 053002. arXiv:2108.05704. Bibcode:2021Metro..58e3002Q. doi:10.1088/1681-7575/ac023f. S2CID 236547235.
• Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)". Metrologia. 58 (5): 052001. Bibcode:2021Metro..58e2001L. doi:10.1088/1681-7575/abe0fc. S2CID 234036217.
• Mohr, Peter J.; Shirley, Eric; Phillips, William D.; Trott, Michael (2022). "On the dimension of angles and their units". arXiv:2203.12392 [physics.gen-ph].

External links

• Media related to Radian at Wikimedia Commons