What is the Meaning of 'Regular Polyhedron' in Mathematics?

What is polyhedron?

A polyhedron can be defined as a 3D-shape having flat faces, straight edges, and sharp vertices (corners). The word ‘polyhedron’ comes form the Greek word, where ‘poly’ means ‘many’ and hedron means ‘surface’. Hence, when many flat surfaces combine together, they form a polyhedron.

Polyhedron Definition

A polyhedron can be defined as a three-dimensional solid made up of polygons having flat surfaces, straight edges, and vertices. Some examples of polyhedron are prism, cube, pyramid, etc., whereas cones, spheres, and cylinders are non-polyhedrons because their sides are not polygons. Their surfaces are curved. The plural form of polyhedron is known as polyhedra. They are divided as prisms, pyramids and platonic solids. Some examples are triangular prism, square prism, rectangular pyramid, square pyramid, and cube (platonic solid).

The following figure shows the various types of polyhedrons.

Counting faces, vertices and edges

The dimensions of a polyhedron are divided as faces, edges, and vertices.

Face: The flat surface of a polyhedron is known as its face.
Edge: The point at which the two faces meet is called the edge.
Vertices: The point at which two edges intersect is known as vertex.

The following figure shows the face, vertex, and edges of a shape.

Polyhedron formula

There exists a relationship between the number of faces, edges, and vertices in a polyhedron. This relationship can be represented as a mathematics formula known as the Euler’s Formula.

Euler’s formula = F + V – E = 2, where, F = number of faces, V= number of vertices and E= number of edges.

By applying the Euler’s formula, the missing part of a polyhedron can be easily found out. It can also be verified whether a polyhedron with the given number of parts exists or not such that a cube has 6 faces, 8 vertices (corner points) and 12 edges. Now let us find out whether a cube is a polyhedron or not by using the Euler’s formula.

F = 6

V = 8

E = 12

Euler’s formula = F + V – E = 2

Where,

F= number of faces

V = number of vertices

E = number of edges

Now let us substitute the values in the formula;

6 + 8 – 12 = 2

Thus, it is proved that a cube is a polyhedron.

Types of Polyhedrons

There are two main divisions of polyhedrons. They are regular polyhedron and irregular polyhedron.

Regular Polyhedron

A regular polyhedron is also called as a platonic solid whose faces are regular polygons and are congruent to each other. All the polyhedral angles are equal in a regular polyhedron. There are five types of polyhedrons. They are:

Tetrahedron: It has 4 faces, 6 edges, and 4 vertices. The shape of each face is an equilateral triangle.
Cube: It has 6 faces, 12 edges, and 8 vertices. The shape of each face is a square.
Regular Octahedron: It has 8 faces, 12 edges, and 6 vertices. The shape of each face is a regular pentagon.
Regular Dodecahedron: It has 12 faces, 30 edges, 20 vertices. The shape of each face is a regular pentagon.
Regular icosahedron: it has 20 faces, 30 edges, and 12 vertices. The shape of each face is an equilateral triangle. The following figure shows the different types of regular polyhedrons.

Irregular Polyhedron

An irregular polyhedron can be defined as having irregular polygonal faces which are not congruent to each other and in which the polyhedral angles are not equal.

Convex Polyhedron

A convex polyhedron is similar to that of a convex polygon. When a line segment which joins any two points on the surface of a polyhedron occurs inside the polyhedron, then it is called a convex polyhedron.

Concave Polyhedron

A concave polyhedron is just like a concave polygon. When a line segment which joins any two points on the surface of a polyhedron comes out of the polyhedron, then it is called a concave polyhedron.

Polyhedrons in everyday life