Overview

Discussion

Encyclopedia

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit
SI derived unit
The International System of Units specifies a set of seven base units from which all other units of measurement are formed, by products of the powers of base units. These other units are called SI derived units, for example, the SI derived unit of area is square metre , and of density is...

. The SI unit of solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

The steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane...

.

The radian is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.2 rad" or "1.2c" (the second symbol is often mistaken for a degree: "1.2°"). 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. In the absence of any symbol radians are assumed, and when degrees are meant the symbol °
°
˚ "modifier letter ring above" is a character of the Spacing Modifier Letters range .It is used in the transliteration of Abkhaz to represent the letter ....

is used.

## Definition

Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

subtended
Subtended
In geometry, an angle subtended by an arc, line or other curve is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context. For example, one may speak of the angle subtended by an arc of a circumference when the angle's vertex is a point on the...

by a circular arc
Arc (geometry)
In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

as the length of the arc divided by the radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

of the arc. One radian is the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

subtended
Subtended
In geometry, an angle subtended by an arc, line or other curve is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context. For example, one may speak of the angle subtended by an arc of a circumference when the angle's vertex is a point on the...

at the center of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

by an arc
Arc (geometry)
In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

that is equal in length to the radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

of the circle. More generally, the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

in radians of such 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. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = .
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 2π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

r /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.

## History

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes
Roger Cotes
Roger Cotes FRS was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the Principia, before publication. He also invented the quadrature formulas known as Newton–Cotes formulas and first introduced what is known today as...

in 1714. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure. The idea of measuring angles by the length of the arc was used already by other mathematicians. For example al-Kashi (c. 1400) used so-called diameter parts as units where one diameter part was 1/60 radian and 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
James Thomson (engineer)
right|300px|James Thomson was an engineer and physicist whose reputation is substantial though it is overshadowed by that of his younger brother William Thomson .-Biography:Born in Belfast, he grew up mostly in Glasgow...

(brother of Lord Kelvin) at Queen's College, Belfast
Belfast
Belfast is the capital of and largest city in Northern Ireland. By population, it is the 14th biggest city in the United Kingdom and second biggest on the island of Ireland . It is the seat of the devolved government and legislative Northern Ireland Assembly...

. He used the term as early as 1871, while in 1869, Thomas Muir
Thomas Muir (mathematician)
Sir Thomas Muir FRS was a Scottish mathematician, remembered as an authority on determinants. He was born in Stonebyres in South Lanarkshire, and brought up in the small town of Biggar. At the University of Glasgow he changed his studies from classics to mathematics after advice from the future...

, then of the University of St Andrews
University of St Andrews
The University of St Andrews, informally referred to as "St Andrews", is the oldest university in Scotland and the third oldest in the English-speaking world after Oxford and Cambridge. The university is situated in the town of St Andrews, Fife, on the east coast of Scotland. It was founded between...

### Conversion between radians and degrees

As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π.

For example:

Conversely, to convert from degrees to radians, multiply by π/180.

For example:

Radians can be converted to turns by dividing the number of radians by 2π.

#### Radian to degree conversion derivation

We know that the length of circumference of a circle is given by , where is the radius of the circle.

So, we can very well say that the following equivalent relation is true:

[Since a sweep is need to draw a full circle]

By definition of radian, we can formulate that a full circle represents:

Combining both the above relations we can say:

radians are equal to one turn
Turn (geometry)
A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...

multiply by , and to convert from grads to radians multiply by . For example,

The table shows the conversion of some common angles.
Units | Values
Turns
Turn (geometry)
A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

0
Degrees
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

30° 45° 60° 90° 180° 270° 360°
The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...

0g 50g 100g 200g 300g 400g

Since a turn
Turn (geometry)
A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

is often identified with the circle constant there is essentially no difference between measuring angles in turns and in radians.

In calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

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 function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s are simple and elegant when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

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 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
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

for sin x :

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

Mathematically important relationships between the sine and cosine functions and the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

(see, for example, Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

) 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.

Another way to see the dimensionlessness of the radian is in the series representations of the trigonometric functions, such as the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

for sin x mentioned earlier:

If x had units, then the sum would be meaningless: the linear term x cannot be added to (or have subtracted) the cubic term or the quintic term , etc. Therefore, x must be dimensionless.

Although 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.

## Use in physics

The radian is widely used in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

when angular measurements are required. For example, angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

is typically measured in radians per second (rad/s). One revolution per second is equal to 2π radians per second.

Similarly, angular acceleration
Angular acceleration
Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha .- Mathematical definition :...

For the purpose of dimensional analysis, the units 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 (k·2π) radians, where k is an integer, they are considered in phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

, whilst if the phase difference of two waves is (k·2π + π), where k is an integer, they are considered in antiphase.

Metric prefix
SI prefix
The International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...

es have limited use with radians, and none in mathematics.

There are 2000π milliradians (≈ 6283.185 mrad) in a circle. So a trigonometric milliradian is just under 16283 of a circle. This “real” trigonometric unit of angular measurement of a circle is in use by telescopic sight
Telescopic sight
A telescopic sight, commonly called a scope, is a sighting device that is based on an optical refracting telescope. They are equipped with some form of graphic image pattern mounted in an optically appropriate position in their optical system to give an accurate aiming point...

manufacturers using (stadiametric) rangefinding in reticles.
The divergence
Beam divergence
The beam divergence of an electromagnetic beam is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the electromagnetic beam emerges. The term is relevant only in the "far field", away from any focus of the beam...

of laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

beams is also usually measured in milliradians.

An approximation of the trigonometric milliradian (0.001 rad), known as the (angular) mil
Angular mil
An angular mil, also mil, is a unit of angle. All versions of the angular mil are approximately the same size as a trigonometric milliradian.-History:The milliradian was first identified in the mid nineteenth Century...

, is used by NATO and other military organizations in gun
Gun
A gun is a muzzle or breech-loaded projectile-firing weapon. There are various definitions depending on the nation and branch of service. A "gun" may be distinguished from other firearms in being a crew-served weapon such as a howitzer or mortar, as opposed to a small arm like a rifle or pistol,...

nery and targeting. Based upon an approximation of = 3.2, there are 6400 mils in a circle, so a NATO mil is 16400 of a circle. Other gunnery systems may use a different approximation to .
Being based on the milliradian, the NATO mil corresponds roughly to an error of 1 m at a range of 1000 m (at such small angles, the curvature is negligible).

Smaller units like microradians (μrads) and nanoradians (nrads) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.

## Criticism

The widespread habit of expressing radians in terms of fractions of π has been criticized because it is like using π as a unit. An angle of π radians represents half a circle, which is a rather arbitrary choice of unit. Various mathematicians and scientists have been critical of the custom of expressing radians as fractions of π, and have proposed using other constants instead. A. Eagle proposed a constant representing one quarter-turn. A constant representing 2π has been proposed because a circle is 2π radians around. Robert Palais proposed to use a "pi with three legs" () to denote 1 turn
Turn (geometry)
A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

, while several people independently have proposed to use the Greek letter τ (tau
Tau
Tau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300.The name in English is pronounced , but in modern Greek it is...

• Angular mil
Angular mil
An angular mil, also mil, is a unit of angle. All versions of the angular mil are approximately the same size as a trigonometric milliradian.-History:The milliradian was first identified in the mid nineteenth Century...

- military measurement
• Trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

• Harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

• Angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian...