Euler’s Theorem Convex Polyhedra Vertices + Faces — Edges = 2

While studying building solids with triangles, rectangles, pentagons, and hexagons, I tried to follow the Four Color Map Theorem and found that it worked for convex polyhedra. What is polyhedra? A polyhedron is a solid with flat faces (poly means many, and hedron means faces). A prism is named by its two congruent (equal) bases and its lateral faces are all rectangles. A pyramid is named by its one base and its lateral faces are all triangles that come to one vertex point. If the solid cannot be named prism or pyramid, we name it by its number of faces. So a cube, can also be called a hexahedron because it has six faces, or a rectangular prism. On page four of the pdf, I showed pictures of many polyhedra (the plural of polyhedron), and named the number of faces from 4 to 162.

We then looked at the relationship between the number of vertices (each a vertex), faces, and edges in a polyhedron. Vertex can be thought of as a corner, edge as the side of one of the faces, and faces are the flat elements of the polyhedron. Polyhedra are three dimensional, faces are two dimensional, edges are one dimensional, and vertices are zero dimensional (a location).

The children analyzed several polyhedra naming the number of vertices, faces, and edges and only a handful of Mathlete classes had one student who noticed a pattern, as did the famous mathematician, Leonard Euler: Vertices + Faces - Edges = 2. When we looked at concave polyhedra (at least one diagonal [line between two vertices] is outside of the polyhedron), we found some Euler numbers of 2 but some not at 2. We concluded as Euler did that V + F - E = 2 works for all convex polyhedra. We even looked at cylinders and spheres which have an Euler number of 1.

During the week, I challenged the children to find dozens of polyhedron in their house, school, or online and record the number of vertices, faces, and edges (make sure the Euler Formula works V+F-E=2 if the polyhedra is convex.