units - Classification of plane angle as fundamental or derived

I recently started with my high school studies and the chapter I am stuck at is units and measurements.

I was told about two types of physical quantities and my teacher gave me the following explanations:

Fundamental Physical Quantities:

Physical quantities that are independent of each other and cannot be broken down into smaller physical quantities are Fundamental.

Derived Physical Quantities:

Physical quantities that can be expressed by simply multiplying or dividing fundamental physical quantities are Derived.

Then while studying about radians, our teacher told us that radian is the unit of derived physical quantity plane angle.

When I asked him to express it as a equation of fundamental physical quantities he was not able to do it.

So my question is.. Is plane angle a fundamental physical quantity or is it derived and if it is derived why cannot it be expressed as fundamental quantities or is it some other problem that I don't know.

Answer

Suppose you take a circle of radius $\ell$ and take an arc of length $\ell$ along the circumference, then 1 radian is the angle subtended by the arc:

More generally, if the length of the arc is $\ell$ and the radius of the circle is $r$ then the angle in radians subtended by the arc is $\ell/r$.

So the radian is a derived unit because it is the ratio of two lengths.