The magnetic penetration depth, λ, is a fundamental parameter of the superconducting state and is commonly determined from precision measurements of the magnetic susceptibility, χ, in the Meissner state. However, extracting λ from such measurements requires an accurate treatment of geometry-dependent demagnetization effects, for which analytic results are only available in a limited number of cases. Here we present a finite-element analysis of the magnetic response of three-dimensional superconducting spheroids and cylinders with finite penetration depth. By calculating the magnetic moment and susceptibility, χ(λ), over a wide range of aspect ratios, we quantify the shape-dependent Meissner response and extract effective demagnetization factors in the λ → 0 limit. For small but finite λ, we further determine effective sample dimensions from the initial variation of χ(λ) and propose an improved analytical expression for these. Our results provide a high-precision procedure for determining λ from volume-exclusion based susceptibility measurements and should be directly applicable to the characterization of novel superconducting materials.