Molecular dynamics simulations of small Cu nanoparticles using three different interatomic potentials at rising temperature indicate that small nanoparticles can undergo solid-solid structural transitions through a direct geometrical conversion route. The direct geometrical conversion can happen for cuboctahedral nanoparticles, which turn into an icosahedra shape: one diagonal of the square faces contracts, and the faces are folded along the diagonal to give rise to two equilateral triangles. The transition is a kinetic process that cannot be fully explained through an energetic point of view. It has low activation energy and fast reaction time in the simulations. The transition mechanism is via the transmission of shear waves initiated from the particle surface and does not involve dislocation activity.