Poliedros e o Teorema de Euler
Data
2014-03-21
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Universidade Federal de Goiás
Resumo
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.
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Parreira, José Roberto Penachia - Poliedros e o Teorema de Euler - 2014 - 80 f. - Dissertação - Programa de Pós-graduação em Matemática (IME) - Universidade Federal de Goiás - Goiânia - Goiás - Brasil.