radian

The radian is the standard unit of angular measure in the SI system. An angle is defined to be one radian if the length of the intercepted arc on a circle of radius r equals r. In other words, θ radians subtend arc length s = rθ. A full circle subtends 2π radians, and a straight angle is π radians. Since a degree is 1/360 of a circle, the relation between radians and degrees is degrees = radians × 180/π, and radians = degrees × π/180. One radian is about 57.2958 degrees.

Radians are preferred in many mathematical and scientific contexts because many formulas become simple when angles are measured in radians. For example, arc length s = rθ and the area of a sector A = (1/2)r²θ hold when θ is in radians. The trigonometric functions sin, cos, and tan have derivatives that assume radians, and their Taylor series expansions use radian measures.

The radian is a derived unit in the SI system. It is defined as a ratio of arc length to radius, which makes it technically dimensionless, though it is typically treated as a unit for clarity in calculations and notation. The radian is widely used across mathematics, physics, engineering, computer science, and astronomy, particularly in contexts involving circular motion, waves, and periodic phenomena where angular measures relate directly to arc length and radius.