Current location - Training Enrollment Network - Mathematics courses - How to prove Euler formula?
How to prove Euler formula?
In fact, there are many formulas called Euler formula. But in geometry, Euler's formula refers to the relationship between the number of vertices V, the number of faces F and the number of edges E of a simple polyhedron: V+F-E=2. The geometric bodies we have learned, such as prisms and pyramids, are all simple polyhedrons. There are many ways to prove Euler's formula. Prove 1: gradually reduce the number of edges of polyhedron, analyze V+F-E, and take simple tetrahedron ABCD as an example to prove. 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 surface and turn it into a plane figure, and prove that V+F 1-E= 1. (1) If one edge is removed and one face is reduced, V+F 1-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. Prove 2: Calculate the internal angles of the faces of the polyhedron, and set the number of vertices V, faces F and edges E of the polyhedron. Cut off a face, make it into a plane figure (expansion diagram), and find the sum of all internal angles σ α (1). In the original drawing, find the sum of internal angles with each surface. 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: σ 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. So the sum of the internal angles of a polyhedron: σ α = (v-n) 3600+(n-2)1800+(n-2)1800 = (v-2) 3600. (2) Defined by (65438).