Polyhedron

In geometry, we may come across different shapes such as one-dimensional, two dimensional and three-dimensional. Each of these geometric shapes possesses some special properties and applications. Polyhedron is a three dimensional solid shape with a certain number of faces, edges and vertices. In this article, you will learn polyhedron definition, types, formula and examples in detail.

Table of Contents:

  • Definition
  • Shape
  • Examples
  • Types
  • Formula
  • Faces edges vertices
  • FAQs

Polyhedron Meaning

The word polyhedron has several meanings imperceptibly in geometry and algebraic geometry. In geometry, a polyhedron is just a three-dimensional solid consisting of a group of polygons, ordinarily connected at their edges. In other words, a polyhedron is a three-dimensional variant of the most common polytope, which defines an arbitrary dimension. The plural name of a polyhedron is “polyhedra” or sometimes “polyhedrons”.

The term “polyhedron” is used somewhat differently in algebraic topology. Here, it is defined as a space built from such “building blocks” like line segments, triangles, tetrahedra, and their higher dimensional analogues by “fixing them together” along with their faces. A polyhedron can be observed as an intersection of half-spaces, whereas a polytope is a bounded polyhedron as shown in the figure below.

Polyhedrons

Polyhedron Shape

A three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices is called a polyhedron. The word ‘polyhedron’ originates from two Greek words: poly and hedron. Here, “poly” means many and “hedron” indicates surface. The names of polyhedrons are defined by the number of faces it has. The below table shows the names and shapes of different polyhedrons based on the number of faces.

Number of faces Name of polyhedron Shape
4 Tetrahedron Tetrahedron
5 Pentahedron Pentahedron
6 Hexahedron Hexahedron
7 Heptahedron Heptahedron
8 Octahedron Octahedron
9 Nonahedron Nonahedron
10 Decahedron Decahedron
12 Dodecahedron Dodecahedron
20 Icosahedron Icosahedron

Polyhedron Examples

We can observe (as given in the below figure) several polyhedrons in our daily existence such as Rubik’s cube, dice, Bucky ball, pyramids and so on.

Polyhedron Examples

Diamond is also an example of polyhedron.

Diamond

Polyhedron Types

Polyhedrons are classified into two types based on the edges they have. They are:

  • Regular polyhedron
  • Irregular polyhedron

Let us understand these types of polygons along with the examples here.

Regular Polyhedron

A regular polyhedron is made up of regular polygons, i.e. all the edges are congruent. These solids are also called platonic solids.

Examples: Triangular pyramid and cube

Regular Polyhedron

Irregular polyhedron

An irregular polyhedron is formed by polygons having different shapes where all the elements are not the same. In this case, all the sides of an irregular polyhedron are not congruent.

Examples: Triangular prism and Octagonal prism

Irregular polyhedron

Polyhedron Formula

If the number of faces and the vertex of a polyhedron are given, we can find the edges using the polyhedron formula. This formula is also known as ‘Euler’s formula’.

F + V = E + 2

Here,

F = Number of faces of the polyhedron

V = Number of vertices of the polyhedron

E = Number of edges of the polyhedron

If we know any two among F, V and E, we can find the third value.

Polyhedron Faces Edges and Vertices

Every polyhedron has three significant components viz faces edges and vertices.

Faces: The flat surfaces that form a polyhedron are called its faces. These faces are two-dimensional polygons.

Edges: The line segments formed by two regions or two flat surfaces (faces) are known as the edges.

Vertices: The point of intersection of the edges is called the vertex of the polyhedron. A polyhedron can have multiple vertices. These are also known as corners of a polyhedron.

The below figure depicts the faces, edges and vertices of a hexahedron.

Faces edges and vertices of polyhedron

Here,

The number of faces = 6

The number of edges = 12

The number of vertices = 8

Verification using Euler’s formula:

F = 6, E = 12, V = 8

F + V = E + 2

6 + 8 = 12 + 2

14 = 14

In the same way, we can identify the vertices, faces and edges of all the polyhedrons.

Frequently Asked Questions on Polyhedron

What are the 5 regular polyhedrons?

The five regular polyhedra include the following:

Tetrahedron (or pyramid)

Cube

Octahedron

Dodecahedron

Icosahedron

How do you identify a polyhedron?

If the solid contains a certain number of faces, edges and vertices that satisfy the Euler’s formula, we can call it a polyhedron.

How many faces does a polyhedron have?

The number of faces of a polyhedron can be determined based on its shape and name. For example, a tetrahedron contains four faces, whereas a dodecahedron has 12 faces.

Can a polyhedron have 10 faces?

Yes, a polyhedron with 10 faces is called a Decahedron. Example for the polyhedron with ten faces is an Octagonal prism.

What are the two types of a polyhedron?

The two types of polyhedrons are regular and irregular. Platonic solids come under the regular polygons. Irregular polyhedrons include prisms and pyramids.

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