In Theorem 1 of his Principia, Newton showed that if one assumes a central force, Kepler’s area law follows. In this work, we report a study where two different ways of deriving this relation, one geometrical (à la Newton) and the other analytical (using vector calculus), were presented to pre-service physics teachers. The students were asked to formulate the essential arguments and to discuss issues that they did not understand. The main findings consist of students’ understanding of the arguments and their core difficulties, as well as a comparative analysis of their preference for more geometrical or analytical derivations.