At college, my Mathematics teacher explained the formula
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?