The radian is officially equal to the number one, and physics teachers hate it
Wrap a circle's radius around its edge and the angle you mark out is one radian. Since the 2019 update of the SI, the radian has been defined as exactly 1, a pure number. That makes maths tidy and mechanics lessons messy, since radians keep appearing and vanishing inside equations.
An angle in radians is simply arc length divided by radius. A full turn equals the whole circumference over the radius, which is two pi, so two pi radians make 360 degrees and a right angle is half of pi. Because it is the SI's coherent unit for plane and phase angles, mathematicians usually drop the symbol altogether: an angle written without units is assumed to be in radians, and degrees get their own sign.
The official symbol is rad, set by the International Bureau of Weights and Measures and the International Organization for Standardization. In 1909 people also wrote a superscript c for circular measure, a plain r or a superscript R, but those are now rare because they are easily mistaken for a degree sign or a radius.
Since metres divided by metres cancel out, the radian is dimensionless. That creates what the physicist Giacomo Prando calls ghostly appearances and disappearances. When a pulley turns, the load moves by radius times angle, and the radian quietly vanishes from the result; angular velocity carries radians per second on one side of an equation but not on the other. Anthony French called this a perennial problem in teaching mechanics, and Oberhofer judged the usual advice to add or drop radians by convention pedagogically unsatisfying. A 1993 metric committee of American physics teachers recommended showing radians only where another angle unit would change the number, as in angular speed, but not in torque or angular momentum.
From 1936 to 2022, no fewer than twelve researchers suggested making plane angle a base quantity with the radian as its unit. Quincey's review found two routes. Giving radius the unit metres per radian breaks the familiar area formula for a circle. Adding a dimensional constant for angle is logically rigorous but strange, and rewriting so many equations will probably keep it from catching on.
Source: Radian
