An electrostatic eigenmode method that describes the coupling between plasmonic nanoparticles is used to model the optical resonances of the 3D plasmonic ruler. The model provides a mathematical description of the ruler that enables us to identify the key resonance in the scattering spectrum that encodes the location of the central nanorod. The model demonstrates excellent agreement with experimentally measured spectra. We show that the spectra can uniquely encode the horizontal and vertical displacements of the central nanorod. From an understanding of the spatial dependence of the plasmonic coupling between the nanorods, we devise a method for estimating the position of the central nanorod and apply this to experimental data. Our method paves the way toward the use of high-resolution spectra from 3D plasmonic oligomers for structural analysis of single entities such as complex macromolecules, DNA scaffolds, proteins, and peptides.