We know that calculating the volume of a three dimensional sphere can be done in spherical coordinates by using $$\iiint dxdydz\to\iiint drd\theta d\phi |\det J|.$$ $J$ is the Jacobian and the determinant is $r^2 \sin \theta$.
The surface area of the sphere can be similarly found by taking the radius to be constant. This is an intuitive thing to do, since the area is made up of the points that are at fixed $R$: $$A=\iint d\theta d\phi \det J|_R=2\pi R^2\int_0^\pi\sin\theta d\theta=4\pi R^2$$ However, let us now consider the problem of finding the circumference of a circular latitude at fixed azimuthal $\theta=\theta_0$ . I know the intuitive geometric reasoning that gives $$C=R\sin\theta_0\int d\phi=2\pi R\sin\theta_0.$$
How does this fact emerge from the Jacobian? In other words, how can one use the Jacobian to arrive at the form of C? I see the sine term emerge with the constant azimuthal angle, which makes a close analogy to the surface area problem discussed earlier, but the units of the jacobian just don't work into $C$: the jacobian goes like $R^2$, but $C$ needs to be proportional to $R$.
I am asking because I wish to apply these conceptional ideas to solve a different problem in higher dimensions that involves generalizing the change-of-basis integration with certain coordinates fixed.
I would appreciate an answer that addresses this at the level of a lower division undergraduate course!