There is a relationship between the number of vertices v, the number of faces f and the number of edges e of a simple polyhedron. This formula is called Euler formula. This formula describes the unique laws of the number of vertices, faces and edges of a simple polyhedron. Methods 1: Gradually reduce the number of edges of polyhedron (using geometric sketchpad), and analyze V+F-E with a simple tetrahedron ABCD as an example. Remove a face to make it a plane figure. After deformation, the number of vertices v, edges v and remaining faces F 1 of tetrahedron remain unchanged. Therefore, in order to study the relationship between V, E and F, we only need to remove one face and turn it into a plane figure, and prove that V+F 1-E =1(1) removes an edge, and then one face is reduced, and V+F1-e remains unchanged. Remove all faces in turn and become a "tree". (2) Every time an edge is removed from the remaining tree, a vertex is reduced, and V+F 1-E remains unchanged until only one edge remains. In the above process, V+F 1-E remains unchanged, and V+F 1-E= 1, so a removed surface is added, and V+F-E =2. For any simple polyhedron, this method has only one line segment left. Therefore, this formula is correct for any simple polyhedron. Method 2: Calculate the internal angles of each face of the polyhedron, and set the number of vertices V, faces F and edges E of the polyhedron. Cut off a face and make it a plane figure (open drawing), and find the sum of the internal angles of all faces. On the one hand, the sum of internal angles is obtained by using the faces in the original drawing. There are f faces, the number of sides of each face is n 1, n2, …, nF, and the sum of the internal angles of each face is: ∑ α = [(n1-2)1800+(N2-2)1800+…. Let a cutting surface be an N-polygon, and the sum of its internal angles is (n-2 n-2) 1800, then among all V vertices, there are N vertices on the side and V-n vertices in the middle. The sum of the internal angles of the V-n vertices in the middle is (V-N) 3600, and the sum of the internal angles of the N vertices on the side is (N-2 n-2) 1800. Therefore, the sum of the internal angles of each facet of a polyhedron: ∑α = (v-n) 3600+(n-2)1800+(n-2)1800 = (v-2) 3600. (2) Defined by (65438).

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