Poliedros de Platão e de Arquimedes : um estudo sobre poliedros clássicos e uma proposta de ensino desses objetos

Abstract

This work aims to study polyhedra, with emphasis on the polyhedra of Plato and Archimedes, dealing with metric relationships and relationships between their elements. Initially, we precisely define what a polyhedron is, the area and volume of a polyhedron and present prisms and pyramids as the main examples of these objects seen in basic education. Furthermore, we deal with the Euler characteristic of a polyhedron and present the proof that V − A + F = 2 for convex polyhedra. Next, we present Plato’s polyhedra (or regular polyhedra), proving that there are only five Plato’s polyhedra and, in addition, we discuss the elements and metric relationships of these polyhedra: number of vertices, edges and faces; the total area and volume of each of the polyhedra with respect to the radii of the circumscribed spheres. We then describe Archimedes’ polyhedra and provide proof of the existence of thirteen, and only thirteen, polyhedra of this class. We also carry out a study of its elements, areas and volumes. Finally, aiming to highlight the contribution of this work to basic education, we report what the BNCC and the PCNs bring regarding the topic and we bring some entrance exam questions involving the topics covered in this dissertation, we present a way of creating tools in GEOGEBRA to do the truncation type 1 and type 2 and we present an educational resource, where we use games, puzzles and concrete material to study the polyhedra of Plato and Archimedes.