This work focuses on a 2D finite volume scheme for the M1 model on conical meshes. The unknowns are the radiative energy E(t, x) and the radiative flux F(t, x). Moreover, when the characteristic parameter of the model vanishes, the flux F tends to 0 and the energy E converges toward the solution of a diffusion equation. For application purposes, it is mandatory for a discretisation of the M 1 model to be consistent with this limit. Such a scheme is called asymptotic preserving (AP). An AP scheme was developed for polygonal meshes. The current work is an extension of this method to conical meshes, whose edges are rational quadratic Bézier curves.