Classification of plane angle as fundamental or derived

Question

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 1

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.

Answer 2

Plane angle is indeed an independent physical quantity--and has been conceptualised as such for millennia, as summarised in Euclid's "Elements" in the third century BCE. The "well-known" relationship between a circular arc and the circle's radius and corresponding central angle:

arc-length = (radius)x("angle in radians")

is confusing because of the misleading interpretation of "angle in radians" as an "angle" expressed in radians: (numerical value) rad. For dimensional consistency, "angle in radians" must have the dimension of length/length = "1"--i.e. the "angle" appears to be dimensionless. This does NOT mean that a radian is a (dimensionless) derived unit--[the SI defines it as the derived unit m/m (= 1)].

What "angle in radians" really means is the numerical value of the (independent physical quantity) ANGLE when the angle is expressed in radians. Thus, if theta is a central angle (with independent dimension A), the numerical value of theta (when theta is expressed in radians) is theta/rad--or the number of radians in theta.

The above relationship between arc-length, radius and central angle theta then becomes:

arc-length = (radius)x(theta/rad)

When rearranged explicitly for theta:

theta = (arc-length)/(radius) rad

we see that theta and rad both have the (independent) dimension A. And the equation is dimensionally consistent: A = (L/L)A = A. [The SI interpretation of "radian" is "the numerical value of one radian when the radian is expressed in radians"--or "the number of radians in one radian." Clearly, this is identically equal to 1.]

BP Leonard

Emeritus Professor of Mechanical Engineering

The University of Akron