Radian

A radian is a unit of angle created when you move the same length round the outside of the circle as the radius. One radius-length around the outside is one radian of angle.

Image by Lucas V. Barbosa

The everyday unit of angle, a degree, measures the amount of rotation. The degree is just a convenient unit of rotation.

To measure distance, for example around the circle, we need to use a different unit, for example the metre.

So when we want to describe one in terms of the other, we need to express it in two units, for example in metres per degree. And if we want velocity, we need three units, for example metres per degree per second.

If I measure the angle of rotation in units of radii travelled, instead of in degrees, I would remove one unit of measurement from my description. One “radian” would tell me both:

• how much I have rotated (because there are 2π of radians to turn in a circle)
• and how far I have travelled (because one radian of rotation is one radius of travel)

I could express my motion as one radian per hour. This tells me both the distance travelled and the angle rotated.

The radian is not a normal unit. It was only formalised in the 1870’s and is classified as a derived standard unit.

Another way to consider it is the following:

• the metre is the SI unit of distance
• metres around the circumference divided by metres of radius is a simple number – a proportion of one expressed in terms of the other
• the ratio of the distance around (circumference) to the distance “in” (radius) is all that is required to define the angle.

One advantage of measuring angle in radians is that it reduces the number of different types of unit required to describe motion. Another is that it includes the unknowable number π. The relationship between rotation and distance must always involve π, making any number approximate and complicated. If I express the distance or angle of the circle in terms of radius, then π is already included. This simplifies expressions that use or imply π, like sine and cosine.