In Regge calculus, space-time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show, such a formulation allows us to replace the length variables by three- or four-dimensional dihedral angles as basic variables. Moreover, we will introduce a first-order formulation, which, in contrast to using flat simplices, does not require any constraints. These considerations could be useful for the construction of quantum gravity models with a cosmological constant.