An analytical solution is derived for the bifurcations of an elastic disc that is constrained on the boundary with an isoperimetric Cosserat coating. The latter is treated as an elastic circular rod, either perfectly or partially bonded (with a slip interface in the latter case) and is subjected to three different types of uniformly distributed radial loads (including hydrostatic pressure). The proposed solution technique employs complex potentials to treat the disc’s interior and incremental Lagrangian equations to describe the prestressed elastic rod modelling the coating. The bifurcations of the disc occur with modes characterized by different circumferential wavenumbers, ranging between ovalization and high-order waviness, as a function of the ratio between the elastic stiffness of the disc and the bending stiffness of its coating. The presented results find applications in various fields, such as coated fibres, mechanical rollers, and the growth and morphogenesis of plants and fruits.