Polyhedron | Meaning, Shapes, Formula, and Examples

A polyhedron is a 3D solid made up of flat polygonal faces, with edges meeting at vertices. Each face is a polygon, and the edges connect the faces at their vertices. Examples include cubes, prisms, and pyramids.
Shapes like cones and spheres are not polyhedrons because they lack polygonal faces.

Polyhedrons can have any polygonal face (triangle, square, pentagon, etc.) and follow Euler's formula.

Polyhedron Examples

There are various examples of polyhedrons, some of the most common examples are listed in the following table:

Polyhedrons	Characteristics	Shape or Form
Cube	Polyhedron Composed of Squares Faces: 6 Vertices: 8 Edges: 12
Tetrahedron	Polyhedron Composed of Equilateral Triangles Faces: 4 Vertices: 4 Edges: 6
Octahedron	Polyhedron Composed of Equilateral Triangles Faces: 8 Vertices: 6 Edges: 12
Dodecahedron	Polyhedron Composed of Regular Pentagons Faces: 12 Vertices: 20 Edges: 30
Icosahedron	Polyhedron Composed of Equilateral Triangles Faces: 20 Vertices: 12 Edges: 30

Real-Life Examples of Polyhedrons

The following illustration contains some real-life examples of polyhedrons:

Polyhedrons Faces, Edges, and Vertices

• Faces: The flat, two-dimensional polygons that make up the polyhedron's surface are known as faces.
• Edges: The edges of a polyhedron are the segments of a straight line that connect two faces. They define the boundaries or points where the faces converge.
• Vertices: Vertices are the polyhedron's corners or meeting points for multiple edges.

Read More: Vertices, Faces, and Edges.

Prisms, Pyramids, and Platonic Solids

Prisms

Prisms are polyhedrons with two parallelogram-shaped lateral faces connecting two congruent polygonal bases. They can be found as triangular, rectangular, or pentagonal prisms, among other shapes. Prisms are frequently found in commonplace items like buildings and packaging.

• Triangular Prism: It has triangular bases and three rectangular lateral faces( faces of a polyhedron that are not based).
• Rectangular Prism: It has rectangular bases and four rectangular lateral faces.
• Pentagonal Prism: It has pentagonal bases and five rectangular lateral faces.

Pyramids

Pyramids are polyhedrons with triangular faces that converge at a single vertex known as the apex along with a polygonal base. Tetrahedrons, square pyramids, and pentagonal pyramids are a few examples of pyramid shapes. Pyramids have been used in construction, including the Egyptian pyramids, and are frequently related to past civilizations.

• Tetrahedron: It has three triangle faces that converge at the top.
• Square Pyramid: Four triangular faces that converge at the top and have a square base.
• Pentagonal Pyramid: This structure has five triangular faces that converge into a pentagonal base.

Platonic Solids

Five convex polyhedrons with identical regular polygonal faces and equal angles make up a distinctive category called "Platonic solids." They consist of the cube, octahedron, dodecahedron, and icosahedron, as well as the tetrahedron. Mathematicians and philosophers have been attracted to the unique symmetry characteristics of platonic solids for centuries. They are related to the philosophical elements of Plato and are seen as depicted geometric forms.

People Also Read:

• Platonic Solids
• Platonic Solid Formula
• Prism Formula

Polyhedron Types

Polyhedrons can be classified into various categories, based on various parameters.

• Based on Edge Length 
  • Regular Polyhedron
  • Irregular Polyhedron

• Based on the Surface Diagonal
  • Convex Polyhedron
  • Concave Polyhedron

Let's understand these types in detail as follows:

Regular Polyhedron

A regular polyhedron is one whose edges are of the same length and is made up of regular polygons. It is a three-dimensional object with sharp vertices and flat faces made of straight edges. These polyhedrons are commonly known as Platonic solids.
The arrangement of vertices, edges, and faces in regular polyhedrons demonstrates symmetry, and the faces are congruent regular polygons.

Some common examples of regular polyhedrons are tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons.

Irregular Polyhedron

Polyhedrons that don't fit into the criteria of regularity are called irregular polyhedrons. Their vertices, edges, and faces are not symmetrically arranged, and they do not all have congruent or regular polygonal faces.
Irregular polyhedrons can have faces of various sizes and forms, as well as variable edge and vertices combinations.

Some common examples of irregular polyhedrons are Cuboid, Irregular Dodecahedrons, and Irregular Icosahedrons.

Convex Polyhedron

Every line segment joining any two points inside the polyhedron completely resides inside or on the polyhedron's surface in a convex polyhedron. In other terms, it is a polyhedron with convex polygons on each face and flat surfaces throughout.

Properties of Convex Polyhedron:

• All of its faces' inner angles are less than 180 degrees.
• Any two faces' intersections are either empty, share an edge, or have a common vertex.

Examples: regular tetrahedron, cube, octahedron, dodecahedron, icosahedron, etc.

Concave Polyhedron

A concave polyhedron is a particular kind of polyhedron that has at least one concave face, or one with an interior angle higher than 180 degrees.

There are line segments connecting points inside a concave polyhedron that may extend beyond the polyhedron's surface. This indicates that in some areas of the polyhedron, the line segment joining two points does not wholly lie inside or on the polyhedron's surface.

Examples: star-shaped polyhedron, Stair-Case-shaped polyhedron.

Some Other Types of Polyhedrons

• Archimedean Solids: Archimedean solids are those convex polyhedrons that have equal edges but have different types of regular polygonal faces. Some examples of these solids include the truncated icosahedron (soccer ball shape) and the rhombicuboctahedron.

• Johnson Solids: Johnson solids are convex polyhedrons that are not regular or Archimedean. They have faces that are regular polygons, but the arrangement of the faces and vertices is irregular. Examples include the pentagonal pyramid and the elongated square pyramid.

Polyhedral Dice

Special dice known as polyhedral dice are used in board games, role-playing games, and mathematics games. They are generally applied to games to add an element of chance or randomness.

Polyhedral dice, as opposed to traditional six-sided dice (D6), have more than six faces, enabling a greater range of outcomes.

Some Examples of Polyhedral dice are:

• D4: This is a tetrahedron-shaped die with four triangular faces.
• D6: This is the six-sided die most people are familiar with it as we all have played ludo,  snakes, and ladder once in our lifetime.
• D8: This die has eight triangular faces.
• D20: The twenty-sided die has twenty equilateral triangular faces.

Polyhedron Formula

Euler's formula states that for any convex polyhedron, the following equation holds:

Euler's formula for Polyhedron

F + V - E = 2

Where, 

• F is the total number of faces,
• V is the total number of vertices, and
• E is the total number of edges.

Let's consider an example to verify the above formula.

Example: Verify the Euler's Formula for Cube.

Solution:

For a Cube,

F = 6, E = 12, V = 8
Thus, 6 + 8 - 12 = 2

Therefore, the formula states that the above figure is true and convex polyhedron i.e., Cube.