High-resolution images of M87* and Sgr A* turn black-hole shadows into practical probes of strong gravity, making it useful to distinguish between observables that encode the overall angular scale of a shadow and those that remain sensitive to the detailed geometry of the underlying spacetime. We compare the Kerr geometry with a static, axisymmetric quadrupolar benchmark spacetime (the Zipoy-Voorhees or $q$-metric) using quantities tied directly to measurement and defined on the local observer sky. In this work, the $q$-metric is used not as a literal astrophysical black-hole model, but as a controlled nonrotating reference geometry that isolates quadrupolar deformation without frame dragging. For the $q$-metric we derive compact finite-distance expressions for the equatorial photon sphere, the critical impact parameter, and the angular radius seen by a static observer, thereby clarifying the domain and limits of shadow formation: the photon sphere exists for $q>-3/2$ with $r_{\mathrm{ph}}^{(q)}=m(3+2q)$, while the requirement that it lies outside the curvature singularity and that $b_{\mathrm{cr}}(q)$ be real and positive restricts observable static shadows to $q>-1/2$, with the angular size decreasing monotonically as $q$ becomes more negative within this domain. In the case of the Kerr spacetime, we use the standard parametric description of spherical photon orbits to compute the shadow at finite distance and arbitrary inclination. To compare the two models operationally, we define $q(a;r_O,{\vartheta }_O)$ by equating a single, well-specified angular radius on the observer’s sky; we adopt the vertical (top) radius to avoid edge degeneracies. In the far field, this yields a single-valued correspondence within the explored domain governed by the critical impact parameters, $b_{\mathrm{K}}(a,{\vartheta }_O)=b_{\mathrm{cr}}(q)$, and a small–deformation approximation provides accurate estimates at moderate spin. We emphasize that this construction defines a controlled size-level correspondence rather than a full degeneracy of the shadow contour. We identify a regime in which static quadrupolar deformations reproduce the Kerr vertical angular scale and show that simple additional observables—the vertical and horizontal radii, the centroid shift and a basic distortion measure—restore discriminating power within the present contour-level benchmark setting. In this way, the analysis separates what can be matched at the level of a single angular-size observable from what remains specifically sensitive to rotation. The framework offers a systematic route from image-plane data to a controlled comparison between Kerr and a static quadrupolar benchmark geometry at finite distance, helping identify which shadow features are associated with scale matching and which remain specific to rotational asymmetry.

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• Classical black holes
• General relativity
• Gravitational lenses
• Singularities In general relativity