Fluorescence resonance energy transfer (FRET) involving a semiconductor nanoparticle (NP) acting as a donor, attached to multiple acceptors, is becoming a common tool for sensing, biolabeling, and energy transfer applications. Such nanosystems, with dimensions that are in the range of FRET interactions, exhibit unique characteristics that are related to the shape and dimensionality of the particles and to the spatial distribution of the acceptors. Understanding the effect of these parameters is of high importance for describing the FRET process in such systems and for utilizing them for different applications. In order to demonstrate these dimensionality effects, the FRET between CdSe/CdS core/shell NPs with different geometries and dimensionalities and Atto 590 dye molecules acting as multiple acceptors covalently linked to the NP surface is examined. Steady-state emission and temporal decay measurements were performed on the NPs, ranging from spherical to rod-like shaped systems, as a function of acceptor concentration. Changes in the NP geometry, and consequently in the distributions of acceptors, lead to distinctively different FRET behaviors. The results are analyzed using a modified restricted geometries model, which captures the dimensionality of the acceptor distribution and allows extracting the concentration of dye molecules on the surface of the NP for both spherical and elongated NPs. The results obtained from the model are in good agreement with the experimental results. The approach may be useful for following the spatial dynamics of self-assembly and for a wide variety of sensing applications.