Radian

A radian is a unit of measuring angles. It is shown by the symbol "rad"^[1] or, less often, ^c (for circular measure). The radian was once an SI supplementary unit, but was changed to a derived unit in 1995. In a circular sector with a central angle of one radian, the arc length is equal to the length of the radius of the circle it is part of.

Definition of a radian

A radian is a unit of angular measure. A revolution of a circle has an angle of 2π radians.

Usage

Most people who do mathematics or physics use radians, rather than degrees, because some kinds of calculations, mostly in trigonometry and calculus, are simpler when using radians rather than degrees. Thus, most calculations related to angular frequency (such as angular velocity) use radians per second.

People who look through a telescope or sniper scope often use milliradians to describe distances as seen through it.

Conversion

1 radian is equal to about 57.3°. There are 2π radians (about 6.28 radians) in a full revolution.^[2] The formula for converting radians to degrees and vice versa is:^[3]

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

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

or:

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

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

we can also say that:

${\displaystyle x{\mbox{ rad}}=\left({\frac {180x}{\pi }}\right)^{\circ }}$.

Related pages

• Steradian
• Tau, the measure of a full circle in radian
• Unit circle