Radians Definition

Radians are a unit that measures angle as the ratio of the angle’s arc length over the radius of a circle. The (equivalent) symbol is used here in the definition to mean that radians are dimensionless by construction and are radius-invariant. In other words, the size of the circle used to measure the angle doesn’t change the value of the measured angle.

A full rotation in radians is equal to (tau) radians, where is the naturally occurring circle constant defined by any circle’s circumference divided by its radius. Shown below are the first ten digits of the circle constant.


  • See the radian angle system for examples of angles measured in radians and the usage of the system in math.
  • See radian vs. degrees for why radians are the preferred unit for measuring angles in math.