Poliedros e o Teorema de Euler - Parreira, José Roberto Penachia (2014)
This work aims is to demonstrate the Euler's Theorem for polyhedra, given by the equation V A + F = 2, where V; A and F are the numbers of vertices, edges and faces, respectively, the polyhedron. A historical survey of the main characters who contributed to the theme was elaborated. De nitions and properties of polygons and polyhedra were given. The statements were constructed in three distinct ways. The rst by Cauchy, commented by Professor Elon Lages Lima. This statement is valid for any polyhedron homeomorphic to a sphere and has the path planning of the polyhedron withdrawing one of its faces. The second statement was prepared by the professor Zoroastro Azambuja Filho, valid for any convex polyhedron, and its path projection of the polyhedron on a plane and comparison of the internal angles of polygons with projection angles of the polygon faces. The third statements was presented by Legendre, also valid for any convex polyhedron, and its path in the projection of a spherical polyhedron surface. We use the Girard's Formula, the sum of the interior angles of a spherical triangle, to complete the demonstration. This work also suggests methods of applying the proof of Euler's Theorem in the classroom for high school students, and resolution of vestibular exercises involving the subject.