See Radian (band) for the Austrian trio.

## Definition

1 rad = m·m–1 = 1

## Explanation

The radian is useful to distinguish between quantities of different nature but the same dimension. For example angular velocity can be measured in radians per second (rad/s). Retaining the word radian emphasizes that angular velocity is equal to 2π times the rotational frequency.

In practice, the symbol rad is used where appropriate, but the derived unit "1" is generally omitted in combination with a numerical value.

The angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle is one radian. Angle measures in radians are often given without any explicit unit. When a unit is given, sometimes the abbreviation rad is used, sometimes the symbol c (for "circular"). Care must be taken with this symbol since it can be mistaken for the ° (degree) symbol.

An angle measuring 1 radian subtends an arc equal in length to the radius of the circle.

There are 2π (approximately 6.28318531) radians in a complete circle, so:

${\displaystyle 2\pi {\mbox{rad}}=360^{\circ }}$
${\displaystyle 1{\mbox{rad}}={\frac {360^{\circ }}{2\pi }}={\frac {180^{\circ }}{\pi }}\approx 57.29577951^{\circ }}$

or:

${\displaystyle 360^{\circ }=2\pi {\mbox{rad}}}$
${\displaystyle 1^{\circ }={\frac {2\pi }{360}}{\mbox{rad}}={\frac {\pi }{180}}{\mbox{rad}}\approx 0.01745329{\mbox{rad}}}$

More generally, we can say:

${\displaystyle x{\mbox{rad}}=x{\frac {180^{\circ }}{\pi }}}$

If, for example, -1,570796 in radians was given, the corresponding degree value would be:

${\displaystyle -1.570796{\mbox{rad}}=-1.570796\cdot {\frac {180^{\circ }}{\pi }}=-90^{\circ }}$

In calculus, angles must be represented in radians in trigonometric functions, to make identities and results as simple and natural as possible. For example, the use of radians leads to the simple identity

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

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

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

The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995.

For measuring solid angles, see steradian.

## Dimensional analysis

Although the radian is a unit of measure, anything measured in radians is dimensionless. This can be seen easily in that the ratio of an arc's length to its radius is the angle of the arc, measured in radians; yet the quotient of two distances is dimensionless.

Another way to see the dimensionlessness of the radian is in the Taylor series for the trigonometric function ${\displaystyle \sin(x)}$:

${\displaystyle \sin(x)=x-{\frac {x^{3}}{3!}}+\cdots }$

If ${\displaystyle x}$ had units, then the sum would be meaningless; the linear term ${\displaystyle x}$ can not be added to the cubic term ${\displaystyle x^{3}/3!}$, etc. Therefore, ${\displaystyle x}$ must be dimensionless.