Polyhedrons, Types of Prisms and Pyramids, Platonic Solid, Euler’s Formula

Polyhedrons

A polyhedron is a 3-dimensional shape that is formed by polygons. Every polygon in a polyhedron is called a face. The faces meet in a line segment that is called an edge. The point of intersection of three or more edges is called a vertex.

There are two important members of the polyhedron family: prisms and pyramids.

Prisms and pyramids are polyhedrons but sphere, cone and cylinder are not polyhedrons because their faces are not polygons.

Prism

A solid shape having its base and top as identical polygons and side faces as rectangles is called a prism. For example, a cube, a cuboid, etc.

Pyramid

A solid shape having its base as any polygon and side faces as triangles with a common vertex is called a pyramid.

For example, a triangular pyramid (tetrahedron), a square pyramid, etc.

Types of Prisms

Triangular Prism

A prism having its base and top as identical triangles and side faces as rectangles is called a triangular prism. A triangular prism has 5 faces, 6 vertices and 9 edges.

Square Prism

If the base and top of a prism are identical squares, then it is called a square prism.

A square prism has 6 faces, 8 vertices and 12 edges.

Cube

If a square prism has all of its faces as identical squares, then it is called a cube.

A cubic prism or a cube has 6 faces, 8 vertices and 12 edges.

Rectangular Prism (Cuboid)

If the base and top of a prism are identical rectangles, then it is called a rectangular prism or a cuboid. A cuboid has 6 faces, 8 vertices and 12 edges.

Pentagonal Prism

If the base and top of a prism are identical pentagons, then it is called a pentagonal prism. A pentagonal prism has 7 faces, 10 vertices and 15 edges.

Types of Pyramids

All lateral faces of a pyramid are triangles and it is named according to the number of sides of its base.

Triangular Pyramid (Tetrahedron)

If the base of a pyramid is a triangle, then it is called a triangular pyramid. A triangular pyramid is also called a tetrahedron.

A triangular pyramid has 4 faces, 4 vertices and 6 edges.

Square Pyramid

A pyramid with a square base is called a square pyramid. A square pyramid has 5 faces, 5 vertices and 8 edges.

Rectangular Pyramid

If the base of a pyramid is a rectangle, then it is called a rectangular pyramid. It has 5 faces, 5 vertices and 8 edges.

Pentagonal Pyramid

If the base of a pyramid is a pentagon, then it is called a pentagonal pyramid. It has 6 faces, 6 vertices and 10 edges.

Naming a Polyhedron

The following table shows the name given to a polyhedron according to the number of its faces.

Number of faces of the polyhedron	Name of the polyhedron
4	Tetrahedron
5	Pentahedron
6	Hexahedron
7	Heptahedron
8	Octahedron
9	Nonahedron
10	Decahedron

Regular Polyhedron or Platonic Solid

If all the faces of a polyhedron are regular polygons, then it is called a regular polyhedron. There are 5 regular polyhedrons, called as platonic solids. It is important to note that the platonic solids have a name based on the number of sides except a cube.

These are:

Tetrahedron: It is a 4-faced polyhedron such that all the faces are equilateral triangles.

Cube: It is a 6-faced polyhedron whose all the faces are squares.

Octahedron: It is an 8-faced polyhedron whose all the faces are equilateral triangles.

Dodecahedron: It is a 12-faced polyhedron whose all the faces are regular pentagons.

Icosahedron: It is a 20-faced polyhedron whose all the faces are equilateral triangles.

Non-polyhedron Prism

A cylinder is a prism because it has identical circular base and top. But it is not a polyhedron, because its base is not a polygon. It has 3 faces (2 flat and 1 curved) and 2 edges. It has no corner.

Non-polyhedron Pyramid

The base of a cone is not a polygon, therefore, it is a pyramid but not a polyhedron. It has 2 faces (1 flat and 1 curved), 1 edge and 1 vertex.

Euler’s Formula

We can easily count the number of faces, vertices and edges of a polyhedron, if the number of sides of base is less than 10. But if the number of sides of base is greater, then counting faces, vertices and edges becomes difficult. To overcome this difficulty, the great Swiss mathematician Leonard Euler discovered a very important relationship among the number of faces (F), vertices (V) and edges (E) of a polyhedron, called Euler’s formula.

According to Euler’s formula, F + V – E = 2

Example 1: An octahedron has 8 faces and 12 edges. How many vertices does it have?

Solution: In the given octahedron, number of faces (F) = 8 and number of edges (E) = 12.

Let the number of vertices in the given octahedron be V.

Using Euler’s formula, we have F + V – E = 2

8 + V – 12 = 2

V = 2 – 8 + 12

V = 6

Thus, the required number of vertices is 6.

Example 2: An icosahedron has 30 edges and 12 vertices. How many faces does it have?

Solution: In the given icosahedron, number of vertices (V) = 12 and number of edges (E) = 30.

Let the number of faces in the given icosahedron be F.