From Richard Haberman's Applied Partial Differential Equations with Fourier Series and Boundary Problems, 4th Edition, page 28, chapter 1.5, in the context of mnemonics to remember the Laplacian in three dimensions for cylindrical coordinates:
"As an aid in minimizing errors, it should be noted that every term in the Laplacian has the dimension of u divided by two spatial dimensions, just as in Cartesian coordinates. Since $\theta$ is measured in radians, this remark aids in remembering to divide $\partial f / \partial \theta$ by $r^2$."
I have tried looking up various other notes and books to no avail; they give the formula as a reference, use a cylindrical volume element to reason about it, appeal to the curvilinear coordinates formula, etc.
I do not know how to even get started on interpreting the two sentences from page 28 I quoted (maybe other than theta being dimensionless as an isolated fact that I cannot relate to other parts of the text). Could you please help me? Any help would be greatly appreciated.