Radian

Definition

The natural unit for measuring angles in physics — based on the circle itself rather than the arbitrary 360 degrees. A full circle is 2π radians.

$\theta = \frac{s}{r},\quad 2\pi\ \mathrm{rad} = 360^\circ$

Symbol: $rad$ · Unit: radian (rad)

Technical definition

The SI unit of plane angle: the angle subtended by an arc whose length equals the radius, so θ = s/r. One full revolution equals 2π radians; calculus formulas for circular motion and oscillations assume radian measure.

Example

The arc-length formula s = rθ only works with θ in radians: a 0.5 m radius wheel turning through 2 radians rolls 1.0 m. Angular velocity in physics is always rad/s, never degrees/s.

Common confusion

Forgetting to switch the calculator to radian mode is one of the most common physics errors — sin(π/2) is 1 in radians but sin(1.5708°) ≈ 0.027 in degree mode. Whenever a formula came from calculus (arc length, angular motion, small-angle approximations), the angle must be in radians.