Teorema de Euler pela Geometria Esférica

Abstract

From basic concepts of plane geometry and spatial geometry we propose a demonstration of Euler's theorem that is accessible, so to speak, to initiated people. I will make a proof that, so far, wasn t found in any high school book that I had access to.Fascinated by its simplicity and perplexed for never have had access to this type of approach when in secondary school, I decided to expose here a material that can help as a consultation and inspiration for teachers seeking new proposals for their spatial geometry classes, in special spherical geometry and polyhedron.Initially we will talk about some concepts of spherical geometry and the results found in spherical triangle that will serve as the basis for demonstrating the Euler relation for convex polyhedra. Then there is a discussion of regularpolyhedra(or Platonic Solids), where we will do a study that shows how many and what are these solids. This result comes from a very simple analysis of polyhedra and their conditions of existence.To illustrate the results developed in this work, I propose, in the end, some activities that can be applied in classrooms for high school students.