In mathematics and physics, the radian is a unit of angle measure. It is the SI derived unit of angle. It is defined as the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. Angle measures in radians are often given without any explicit unit. When a unit is given, it is rad.
There are 2π (about 6.283185) radians in a complete circle, so:
2π rad = 360°
1 rad = 360/(2π)° = 180/π° (approximately 57.29578°).
360° = 2π rad
1° = 2π/360 rad = π/180 rad
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
which is the basis of many elegant identities in mathematics.
For measuring solid angles, see steradian.
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.
If x had units, then the sum would be meaningless; the linear term x can not be added to the cubic term x3 / 3!, etc. Therefore, x must be dimensionless.