# Euler's formula relates the number of faces, edges, and vertices in a polyhedron

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

1. 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.
2. Show that $F-E+V=2$ for above three dimensional figures.