In the last decades, developments in engineering and material sciences have enabled the possibility of manufacturing beam-like structures with variable cross-section, for instance via 3D printing, to achieve enhanced mechanical properties. In this context, we investigate the buckling behaviour of beams with variable inertia where the inertia changes in a sinusoidal way. In particular, we compute how the buckling load changes as a function of the amplitude and the frequency of the inertia. We compare the results predicted by two different planar beam theories (i.e., a linearized inextensible beam theory and a linearized extensible beam theory) via Finite Element Method. As expected, the buckling load depends on the sinusoidal form of the inertia. In particular, when the number of oscillations of the inertia grows, the critical load reaches a plateau value equal to the Euler critical load calculated with an equivalent inertia equal to the square root of the product between the inertia maximum and minimum values.