Radian

The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the center of a plane circle by an arc that is equal in length to the radius. The unit is defined in the SI as the coherent unit for plane angle, as well as for phase angle. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.

Definition

One radian is defined as the angle at the center of a circle in a plane that is subtended by an arc whose length equals the radius of the circle. 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 magnitude in radians of the subtended angle, s is arc length, and r is radius. A right angle is exactly π/2 radians.

The angle corresponding to one complete revolution is the length of the circumference divided by the radius, which is 2πr/r, or 2π. Thus, 2π radians is equal to 360 degrees. The relation 2π rad = 360° can be derived using the formula for arc length, ⁠

ℓ

arc

=

2

π

r

(

θ

360

∘

)

{\displaystyle \textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}

⁠.