radians

Radians are the standard unit of angular measure in the International System of Units (SI). A radian is defined so that an angle subtending an arc length equal to the circle’s radius on a circle of radius r measures 1 radian. From this definition, a full circle corresponds to 2π radians, which is also 360 degrees. Therefore, 1 radian is approximately 57.2958 degrees. The relationship between radians and degrees is given by degrees = radians × 180/π and radians = degrees × π/180.

A convenient way to think about radians is as a ratio: θ radians = arc length s divided by radius r (θ = s/r). This makes arc length an immediate measure of angle: s = rθ. The central angle in radians is simply the arc length divided by the radius, and a full circle has θ = 2π.

As a unit, the radian is technically dimensionless because it is a ratio of two lengths, yet it is treated as a unit (rad). Using radians simplifies many mathematical formulas: the derivatives and integrals of trigonometric functions assume angles are in radians, and series expansions for functions such as sin, cos, and exp are most natural in radians.

Applications of radians span mathematics, physics, engineering, and computer science, where angular measures, rotational motion, and periodic phenomena are analyzed.