Suppose you have a right circular conical frustum with bottom base radius $r_1$ and top base radius $r_2$, and height $h$.
Now you take a (perspective) image of this frustum. The camera used to create the image has a known focal length, but its position and orientation are unknown.
Is it possible to determine from the image the relative dimensions of the frustum, i.e. the ratio $\dfrac{r_2}{r_1}$ and the ratio $\dfrac{h}{r_1}$ ?
It is assumed that the equations of the ellipses (in the image) corresponding to the top and bottom bases are known.
"Is it possible...?" Short answer: no.
Not so short answer:
the focal length of the used lens does not matter, perspective is only a matter of point of view (here in the physical sense of the words). But, position and orientation are unknown, as you say.
You would need
i) more data, e.g. the distance to the top, the distance to the bottom to get a ratio of the different reduction/blowup (which must take into account the post-processing of the photo).
ii) to guess many assumptions, e.g. the orientation of the frustum, as there are multiple ways to get the same ellipses in the image. Next, what makes you sure the bases of your frustum are circles and parallel planes?
iii) or a second photo with known displacement of the camera. This would enable to get the third dimension which is missing in a single (flat) photo.
For more see Photogrammetry.