Despite decades of studying the Sun, the coronal heating problem remains unsolved. One fundamental issue is that we do not know the spatial scale of the coronal heating mechanism. At a spatial resolution of 1000 km or more, it is likely that most observations represent superpositions of multiple unresolved structures. In this Letter, we use a combination of spectroscopic data from the Hinode EUV Imaging Spectrometer and high-resolution images from the Atmospheric Imaging Assembly on the Solar Dynamics Observatory to determine the spatial scales of coronal loops. We use density measurements to construct multi-thread models of the observed loops and confirm these models using the higher spatial resolution imaging data. The results allow us to set constraints on the number of threads needed to reproduce a particular loop structure. We demonstrate that in several cases million degree loops are revealed to be single monolithic structures that are fully spatially resolved by current instruments. The majority of loops, however, must be composed of a number of finer, unresolved threads, but the models suggest that even for these loops the number of threads could be small, implying that they are also close to being resolved. These results challenge heating models of loops based on the reconnection of braided magnetic fields in the corona.