Euler's formula relates the number of faces, edges, and vertices in a polyhedron
By Eulers formula relates number of faces (F ), edges (E), and vertices (V) of a polyhedron (a solid with flat faces). Euler's formula is given by \begin{align} F-E+V=2. \end{align}
Let us check Euler's formula on a few polyhedron.
Cube
A cube is a polyhedron as it has flat faces. A face is the flat part of the polyhedron. A cube has 6 faces.
The intersection two faces is called an edge. It is a line. A cube has 12 edges.
The intersection of three (or more) edges is called a vertex. These are the corner points. A cube has 8 vertices.
A cube has 6 faces ($F=6$), 12 edges ($E=12$) and 8 vertices ($V=8$). Thus, \begin{align} F-E+V=6-12+8=2. \end{align} The number of faces, edges and vertices in a cube satisfies the Euler's formula.
A solid bounded by six rectangular plane faces is called a cuboid. A book, a brick, or a matchbox are examples of cuboid.
A cuboid whose length, breadth and height are equal is called a cube.
Prism
A solid whose two faces are parallel polygons and the side faces are rectangular is called a prism.
If polygon in a prism is a triangle then it is called a triangular prism. The prism you see in optics lab is a triangular prism.
A triangular prism has 2 triangular parallel faces and 3 rectangular faces. The total number of faces in a triangular prism is $F=2+3=5$. It has 9 edges and 6 vertices. Thus, \begin{align} F-E+V=5-9+6=2. \end{align} The number of faces, edges and vertices in a triangular prism satisfies the Euler's formula.
If polygon in a prism is a rectangle then we get a cuboid. A cuboid is also called a rectangular prism.
Pyramid
A pyramid is a solid whose base is a polygon and whose side faces are triangles having a common vertex. The common vertex is called vertex of the pyramid.
If base of a pyramid is a square then we get a square pyramid. If base is a rectangle then we get a rectangular pyramid. A square (or rectangular) pyramid has 5 faces (1 base and 4 triangular side faces), 8 edges and 5 vertices. Thus, \begin{align} F-E+V=5-8+5=2. \end{align} The number of faces, edges and vertices in a rectangular pyramid satisfies the Euler's formula.
If base of a pyramid is a triangle then we get triangular pyramid. A triangular pyramid is also called tetrahedron. It has 4 faces, 6 edges and 4 vertices. Thus, \begin{align} F-E+V=4-6+4=2. \end{align} The number of faces, edges and vertices in a tetrahedron satisfies the Euler's formula.
The number of faces, edges and vertices in common polyhedron is given in the following table.
| Name | Faces (F) | Edges (E) | Vertices (V) |
|---|---|---|---|
| Cuboid (or Cube) | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square (or rectangular) pyramid | 5 | 8 | 5 |
| Triangular pyramid (tetrahedron) | 4 | 6 | 4 |
The Euler's formula $F-E+V=2$ is called a topological invariance.
Exercise
- Write the number of vertices (V), faces (F) and edges (E) in (a) cuboid (b) cube (c) prism (d) square pyramid (e) triangular pyramid - tetrahedron.
- Show that $F-E+V=2$ for above three dimensional figures.
