How do you verify Euler's theorem?

Answer 1

If a planar graph G is drawn in the plane, so that no two edges cross, the plane is divided into a number of regions which may be called "faces". Euler's Theorem (for planar graphs): Let G be a connected planar graph drwawn in the plane. If there are v vertices, e edges, and f faces, then v - e + f = 2. An application of this theorem gives Euler's Theorem for polyhedra.

Euler's Theorem (for polyhedra): If a convex polyhedron has v vertices, e edges, and f faces, then v - e + f = 2 For particular polyhedra is easy to confirm the result stated in theorem. For example, a cube has 8 vertices (v = 8), 12 edges (e = 12), and 4 faces (f = 4) So, v - e + f = 8 - 12 + 4 = 2.A tetrahedrom has v = 4, e = 6, and f = 4. So, v - e + f = 4 - 6 + 4 = 2. Look at this site to understand better (you can see pictures there).

http://www.ics.uci.edu/~eppstein/junkyard/euler/ Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V-E+F=2 The polyhedron formula, of course, can be generalized in many important ways, some using methods described below. One important generalization is to planar graphs. To form a planar graph from a polyhedron, place a light source near one face of the polyhedron, and a plane on the other side. The shadows of the polyhedron edges form a planar graph, embedded in such a way that the edges are straight line segments. The faces of the polyhedron correspond to convex polygons that are faces of the embedding. The face nearest the light source corresponds to the outside face of the embedding, which is also convex. Conversely, any planar graph with certain connectivity properties comes from a polyhedron in this way.

You can see here that there are 8 vertices, 14 edges, and 8 faces. So,

v - e + f = 8 - 14 + 8 = 2 Look at this site also.

http://www.highpointsmath.com/SiteMap/Polyhedron.html

Polyhedron * A polyhedron is a space figure each of whose faces is a polygon. * In other words, a polyhedron is a solid shape whose faces are all polygons. * Cubes, prisms, and pyramids are polyhedra. More about Polyhedron * A regular polyhedron is a polyhedron in which all faces are regular polygons of the same shape and size. Name the polyhedron that has 4 faces, 6 edges, and 4 vertices. Choices: A. hexahedron B. cone C. tetrahedron D. octahedron Correct Answer: C Solution: Step 1:A tetrahedron or a triangular pyramid is a pyramid with a triangular base.Step 2: The net of a tetrahedron that can be folded and joined to form a tetrahedron is as shown.Step 3:The points 2, 3, and 4 forms the base, and the sides join at point 1 to form a pyramid. Step 4: There are 4 faces, 6 edges, and 4 vertices in a tetrahedron. Step 5: So, tetrahedron is a polyhedron that has 4 faces, 6 edges, and 4 vertices.