Radian

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 ${\displaystyle {\text{rad}}}$, 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 value, and 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.^[2] 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. Conversely, the length of the intercepted arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ.

As the ratio of two lengths, the radian is a pure number.^[a] In SI, the radian is defined as having the value 1.^[6] As a consequence, in mathematical writing, the symbol "rad" is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used.

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, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π ≈ 57.295779513082320876 degrees.^[7]

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.

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]

Unit symbol

The International Bureau of Weights and Measures^[16] and International Organization for Standardization^[17] 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,^[18] 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.

Conversions

A chart to convert between degrees and radians