The radian, denoted by the symbol rad, is the dimensionless unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the center of a plane circle by an arc that is equal in length to the radius.: "One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius." The unit is defined in the SI as the coherent unit for plane angle, as well as for phase angle.: "The radian is the coherent unit for plane angle. ... The radian is also the coherent unit for phase angle." Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.
Definition
One radian is defined as the angle at the center of a circle in a plane that is subtended by an arc whose length equals 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, , where is the magnitude in radians of the subtended angle, is arc length, and is radius. A right angle is exactly radians.
The angle corresponding to one complete revolution is the length of the circumference divided by the radius, which is , or . Thus, radians is equal to 360 degrees. The relation can be derived using the formula for arc length, . Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle, . This can be further simplified to . Multiplying both sides by gives .
Unit symbol The International Bureau of Weights and Measures and International Organization for Standardization specify rad as the symbol for the radian. Alternative symbols that were in use in 1909 are c (the superscript letter c, for "circular measure"), the letter r, or a superscript , but these variants are infrequently used, as they may be mistaken for a degree symbol (°) or a radius (r). Hence an angle of 1.2 radians would be written today as 1.2 rad; archaic notations include 1.2 r, 1.2, 1.2, or 1.2.
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.
Dimensional analysis
Plane angle may be defined as , where is the magnitude in radians of the subtended angle, is circular arc length, and is radius. One radian corresponds to the angle for which , hence .: "One radian corresponds to the angle for which " However, 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 gives 1 radian as 1 m2/m2 = 1.: "Also, as alluded to in , the radian can be defined in terms of the area A of a sector (), in which case it has the units m2⋅m−2." The key fact is that the radian is a dimensionless unit equal to 1. In SI 2019, the SI radian is defined accordingly as .: "One radian corresponds to the angle for which , thus ." It is a long-established practice in mathematics and across all areas of science to make use of .
Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by centimetres, where is the magnitude of the radius of the pulley in centimetres and is the magnitude of the angle through which the pulley turns in radians. When multiplying by , the unit radian does not appear in the product, nor does the unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the angular velocity of a rolling wheel, , radians appear in the units of but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". 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".
In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s2), and torsional stiffness (N⋅m/rad), and not in the quantities of torque (N⋅m) and angular momentum (kg⋅m2/s).
At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle".; ; ; ; ; ; ; ; ; ; 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, . 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". 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.
In particular, Quincey identifies Torrens' proposal to introduce a constant 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, , is modified to become , and the Taylor series for the sine of an angle becomes:
where is the angle in radians. The capitalized function is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while is the traditional function on pure numbers which assumes its argument is a dimensionless number in radians. The capitalised symbol can be denoted if it is clear that the complete form is meant.
Current SI can be considered relative to this framework as a natural unit system where the equation is assumed to hold, or similarly, . This radian convention allows the omission of in mathematical formulas.
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, and Mathematica's unit system similarly considers angles to have an angle dimension.
Conversions
Between degrees As stated, one radian is equal to . Thus, to convert from radians to degrees, multiply by .
For example:
Conversely, to convert from degrees to radians, multiply by .
For example:
Radians can be converted to turns (one turn is the angle corresponding to a revolution) by dividing the number of radians by 2.
Between gradians
One revolution corresponds to an angle of radians, which equals one turn, and to 400 gradians (400 gons or 400g). To convert from radians to gradians multiply by , and to convert from gradians to radians multiply by . For example,
Usage
Mathematics
In calculus and most other branches of mathematics beyond practical geometry, angles are measured in radians. This is because radians have a mathematical naturalness that leads to a more elegant formulation of some important results.
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
which is the basis of many other identities in mathematics, including
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 , the evaluation of the integral , and so on). In all such cases, it is appropriate that the arguments of the functions are treated as (dimensionless) numbers—without any reference to angles.
The trigonometric functions of angles also have simple and elegant series expansions when radians are used. For example, when is the angle expressed in radians, the Taylor series for becomes:
If were the angle but expressed in degrees, i.e. , then the series would contain messy factors involving powers of :
In a similar spirit, if angles are involved, 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 angles expressed in radians (and messy otherwise). More generally, in complex-number theory, the arguments of these functions are (dimensionless, possibly complex) numbers—without any reference to physical angles at all.
Physics The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically expressed in the unit radian per second (rad/s). One revolution per second corresponds to 2 radians per second.
Similarly, the unit used for angular acceleration is often radian 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 angle difference of two waves can also be expressed using the radian as the unit. For example, if the phase angle difference of two waves is () radians, where is an integer, they are considered to be in phase, whilst if the phase angle difference of two waves is () radians, with an integer, they are considered to be in antiphase.
A unit of reciprocal radian or inverse radian (rad−1) is involved in derived units such as meter per radian (for angular wavelength) or newton-metre per radian (for torsional stiffness).
Prefixes and variants Metric prefixes for submultiples are used with radians. A milliradian (mrad) is a thousandth of a radian (0.001 rad), i.e. . There are 2 × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.
The angular mil is an approximation of the milliradian used by NATO and other military organizations in gunnery and targeting. Each angular mil represents of a circle and is % 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 ; for example Sweden used the streck and the USSR used . 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 rad (around 4.8481 microradians).
History
Pre-20th century The idea of measuring angles by the length of the arc was in use by mathematicians quite early. For example, al-Kashi (c. 1400) used so-called diameter parts as units, where one diameter part was radian. They also used sexagesimal subunits of the diameter part. Newton in 1672 spoke of "the angular quantity of a body's circular motion", but used it only as a relative measure to develop an astronomical algorithm.