The aim of this work is to show how to implement some of the methods of curved finite elements of class C1 introduced and analyzed in a previous work by the first author. These methods use curved finite elements which are C1-compatible with the Argyris triangle. The careful description of the implementation is completed by some numerical experiments which show the very high degree of accuracy of the associated interpolation. Subsequently, in a second part, we will show the efficiency of such methods for solving thin plate or thin shell problems set on curved boundary domains.