The radian is the SI unit of plane angle. Its symbol is rad. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius, which makes a full turn equal to two pi radians, or about 6.28319.
Because it is defined as a ratio of two lengths, the radian is dimensionless. That is not a technicality but the whole point. It is the reason the derivative of the sine function is the cosine function, with no stray constant attached, and the reason the small-angle approximation says that the sine of a small angle is very nearly the angle itself. Every one of those results acquires an ugly factor of pi over 180 if degrees are used instead.
The name was coined by James Thomson, brother of Lord Kelvin, in an examination question at Queen's College Belfast in 1873, though the concept had been in use without a name for a century. Roger Cotes had described the advantages of measuring angles in this way in work published in 1722.
Every programming language and mathematical library expects radians. The trigonometric functions in C, Python, JavaScript and their descendants all take radian arguments, which is a routine source of bugs when a value arrives from a user interface in degrees. Angular velocity in physics and engineering is expressed in radians per second, and rotational quantities such as torque and angular momentum follow the same convention.
Rotation is measured in radians per second for the same reason. Angular velocity in radians per second multiplied by a radius in metres gives a speed in metres per second directly, with no conversion factor, so a wheel turning at 100 radians per second on a 0.3 metre radius has a rim speed of 30 metres per second. The same coherence appears in the small-angle approximation, which says that for a small angle expressed in radians the sine, the tangent and the angle itself are nearly equal. That approximation underlies the pendulum formula, the analysis of lenses and half the derivations in an engineering textbook, and it holds only when the angle is in radians.
One radian equals about 57.29578 degrees, and a full circle equals two pi radians, roughly 6.28319.
| Unit | Symbol | 1 Radian equals | 1 of these equals | Converter |
|---|---|---|---|---|
| Second of arc | arcsec | 206264.806247 arcsec | 0.0000048481368111 rad | Radians to Seconds of arc |
| Minute of arc | arcmin | 3437.74677078 arcmin | 0.000290888208666 rad | Radians to Minutes of arc |
| Degree | deg | 57.2957795131 deg | 0.0174532925199 rad | Radians to Degrees |