In a paper (posthumously) co-authored by Isaac Newton himself, the primacy of geometric notions in pedagogical expositions of centripetal acceleration has been clearly asserted. In the present paper we demonstrate how this pedagogical prerogative can inform the design of an experiment involving an accelerometer-equipped smartphone rotating uniformly in a horizontal plane. Specifically, the location of the sensor itself within the body of the smartphone will be determined using a technique that is purely geometrical in nature, relying on nothing more than the notion that centripetal accelerations are centrally pointing. The complete absence of algebraic manipulations obliges students to focus exclusively on the development of their geometrical reasoning abilities. In particular, it provides a healthy challenge for those algebraically accomplished students for whom equations, calculations, and data tables represent a means of avoiding a direct confrontation with the imposing spectre of material that is otherwise purely conceptual in nature.