Are radians, degrees and gradians not units?

Question

At college, my Mathematics teacher explained the formula

$$l=r\theta$$

where $r$ is the radius of a circle, $\theta$ is a central angle of the circle and $l$ is the length of the arc subtended by the angle $\theta$.

Then he explained that the units used for $r$ and $l$ in the formula, must be same. So if $r$ is expressed in meters then so is $l$.After this he made a statement that seem quite awkward to me. He said,

$\theta$ is unitless.

He explained his point (after I questioned it) by saying that if $\theta$ has a unit, then the unit of $l$ will be the unit of $r$ multiplied by the unit of $\theta$ and not meters or centimeters or any other unit (which it should be). He said that the terms radians, degrees and gradians are just used to measure the angles, but they are not counted as units. I think that this is false, but I'm not sure of its exact reality.

Is my teacher's statement true? It it isn't, shouldn't the formula be written like this $$l\times1 \ rad=r\theta$$to satisfy the units on both sides?

Answer 1

degrees and radians are units of angle. Angles are not "unitless". Rather, they are "dimensionless" in the sense of dimensional analysis in physics.

Answer 2

One point of dimensional analysis is to ensure our description of the physical world is not dependent on the units of measure employed, i.e. two gentlemen using $cm$ and $m$ for length measurement do not come up with contraddicting statements on, say, angles.

Imagine a circumference is shown, and an angle subtended by a given arc from the center. If we need to define a dimensionally consistent way to measure such angle, it has to be independent from whether lenghts measurements are done in, say, $cm$ or $m$.

By observing that the angle can be characterised by the ratio between subtended arc and circumference (first formula you posted), we conclude the units we use for the angles are dimensionless. The first gentleman will measure the circumference to be $10 m$ long, and the arc $1 m$. The second will meausure the circumference to be $10000 cm$ long, and the arc $1000 cm$. The ratio will be invariant upon choice of length units: hence we say, angle units are dimensionless.

If one applies the same reasoning to areas, for example, and one verifies that units of area have, as expected, units $[L^2]$ , i.e. the measured area by the two observers will change according to a power law with exponent $2$.

Answer 3

A. In addition to base units of measurement (meter, second, ...) one can also use derived units of measurement. They are superfluous, in the sense the derived units are derived from the base ones. But they are not completely interchangeable.

For instance [Joule] is a derived unit for the energy, work and heat. It is derived from base units. Indeed [Joule]=[Newton$\cdot$meter], where [Newton] is a derived unit of the force. The process is recursive. On the contrary the angular momentum, a different physical quantity, is also measured in [Newton$\cdot$meter] but you cannot use [Joule] to measure the angular momentum.

So derived units can be used as an alternative to base units only for prescribed physical quantities.

B. There are physical quantities that are dimensionless, that is their base unit is [$1$]. Now you can apply the same reasoning made at point A. Given prescribed dimensionless physical quantities let's associate a derived unit. For instance, to the ratio of the length of a circular arc by the length of its radius (both measured with the same unit of measurement) let's associate a new derived unit: the [radian]. So when it comes to dimensional analysis all derived units are changed into their corresponding base units and in this case [radian] becomes [$1$], that is, "vanishes" out.

You can apply the same reasoning to all the other dimensionless physical quantities (that is, having base unit [$1$]) and their derived units, like degree, gradian, neper, bel, ...

C. Just to close the example in point A, the angular momentum is given by the derivative of the energy with respect to the angle: $$J=\frac{dE}{d\theta}$$ For what said $J$ does not have a derived unit, but $E$ and $\theta$ do, being [Joule] and [radian] rispectively. So you can say that the angular momentum is measured in [Newton$\cdot$meter] or also in [Joule/radian]