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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 ${\displaystyle {\text{rad}}}$,[1] is the SI unit for measuring angles, 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 subtends; one radian is 180/π degrees or just under 57.3°.[a][2] The unit was formerly an SI supplementary unit (before that category was abolished in 1995) and the radian is now considered an SI derived unit.[3] The radian is defined in the SI as being a dimensionless value, and its symbol is accordingly often omitted, especially in mathematical writing.

The radian, denoted by the symbol ${\displaystyle {\text{rad}}}$,[1] is the SI unit for measuring angles, 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 subtends; one radian is 180/π degrees or just under 57.3°.[a][2] The unit was formerly an SI supplementary unit (before that category was abolished in 1995) and the radian is now considered an SI derived unit.[3] The radian is defined in the SI as being a dimensionless value, and its symbol is accordingly often omitted, especially in mathematical writing.

## Definition

Radian describes the plane angle subtended by a circular 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 that is equal in length to the radius of the circle. More generally, the 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, 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 = .

While it is normally asserted that, as the ratio of two lengths, the radian is a "pure number", although Mohr and Phillips dispute t

Radian describes the plane angle subtended by a circular 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 that is equal in length to the radius of the circle. More generally, the 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, 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 = .

While it is normally asserted that, as the ratio of two lengths, the radian is a "pure number", although Mohr and Phillips dispute this assertion.[4] However, in mathematical writing, the symbol "rad" is almost always omitted.[4] When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used. The radian is defined as 1.[5] There is controversy as to whether it is satisfactory in the SI to consider angles to be dimensionless.[6] This can lead to confusion when considering the units for frequency and the Planck constant.[4][7]

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

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 2πr / r

While it is normally asserted that, as the ratio of two lengths, the radian is a "pure number", although Mohr and Phillips dispute this assertion.[4] However, in mathematical writing, the symbol "rad" is almost always omitted.[4] When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used. The radian is defined as 1.[5] There is controversy as to whether it is satisfactory in the SI to consider angles to be dimensionless.[6] This can lead to confusion when considering the units for frequency and the Planck constant.[4][7]

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 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.[8]

The relation 2π rad = 360° can be derived using the formula for arc length. Taking the formula for arc length, or

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714.[9][10] 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.[11]

The idea of measuring angles by the length of the arc was alr

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.[12]

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.[13][14][15] 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.[16]

The International Bureau of Weights and Measures[17] and International Organization for Standardization[18] 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,[19] 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.

## Conversions

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

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

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 π/180.

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

For example:

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

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

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 ${\displaystyle r}$For example:

${\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}$Conversely, to convert from degrees to radians, multiply by π/180.

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

For example:

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

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

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

#### Radian to degree conversion derivation

The length of circumfere

For example:

${\displaystyle 1^{\circ }=1\cdot {\frac {\pi }{180^{\circ }}}\approx 0.0175{\text{ rad}}}$${\displaystyle 23^{\circ }=23\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 ${\displaystyle r}Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2$π.

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:

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

By the definition of radian, a full circle represents:

Combining both the above relations:

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