Lipid nanodiscs are nanometric bilayer patches enveloped by confining structures, commonly composed of membrane scaffolding proteins (MSPs). To resolve the interplay between MSP geometry, lipid confinement, and membrane material properties on the nanodisc shape, we apply a continuum elastic theory accounting for lipid bending, tilting, and area deformations. The equilibrium nanodisc shape is then determined by minimizing the elastic free energy functional. Analytic expressions derived under simplifying assumptions demonstrate that the nanodisc shape is sensitive to its size, lipid density, and the lipid tilt and thickness imposed at the contact with the MSP. Under matching physical parameters, these expressions quantitatively reproduce the shape of nanodiscs seen in molecular dynamics simulations, but only if lipid tilt is explicitly considered. We further demonstrate how the bending rigidity can be extracted from the membrane shape profile by fitting the numerically minimized full elastic functional to the membrane shape found in simulations. This fitting procedure faithfully informs on the bending rigidity of nanodiscs larger than ca. 5 nm in radius. The fitted profiles accurately reproduce the increase in bending modulus found using real-space fluctuation analysis of simulated nanodiscs and, for large nanodiscs, also accurately resolve its spatial variations. Our study shows how deformations in lipid patches confined in nanodiscs can be well described by a continuum elastic theory and how this fit can be used to determine local material properties from shape analysis of nanodiscs in simulations. This methodology could potentially allow direct determination of lipid properties from experiments, for example cryo-electron microscopy images of lipid nanodiscs, thereby allowing to guide the development of future nanodisc formulations with desired properties.