radian
Angleexact by definition
The radian is the SI coherent unit of plane angle: the angle subtended at a circle's centre by an arc equal in length to the radius. Since it is a ratio of two lengths it is dimensionless, so the conversion factor to SI is exactly 1. A full turn is \(2\pi\) radians, and one radian is about 57.2958 degrees.
Watch out: Every trigonometric identity in calculus assumes radians. The derivative of \(\sin x\) is \(\cos x\) only in radians; in degrees it picks up a factor of \(\pi/180\). Calculator mode errors are almost always a degree/radian mix-up.
| 1 rad | 57.29578 ° |
The name is attributed to James Thomson, who used it in an examination paper at Queen's College Belfast in 1873. The concept is older than the word, having been used implicitly wherever arc length was compared with radius.
About the radian
The radian looks like an inconvenient choice until you differentiate something. Define angle as arc length divided by radius and the small-angle limit \(\sin\theta \approx \theta\) becomes exact as \(\theta \to 0\), which makes \(\frac{d}{dx}\sin x = \cos x\) come out clean. In degrees the same derivative is \(\frac{\pi}{180}\cos x\), and that stray factor propagates through every series expansion, every oscillation solution and every Fourier transform. Radians are not a convention with a nicer feel; they are the only angle measure in which calculus does not carry luggage.
The definition also makes arc length free. An angle \(\theta\) in radians on a circle of radius \(r\) subtends an arc of exactly \(r\theta\), and sweeps an area of \(\tfrac{1}{2}r^2\theta\). A robot arm 0.8 m long rotating 1.2 rad moves its tip 0.96 m, with no conversion. The same identity is why angular velocity in rad/s multiplied by radius gives linear velocity in m/s directly, the relation \(v = \omega r\) that underpins every gear, belt and wheel calculation. Because the radian is a ratio of lengths it is formally dimensionless, so it appears and disappears from unit checks at will, which is either elegant or maddening depending on how the sheet is going.