A polyhedron is a three-dimensional shape with flat polygonal faces, straight edges, and sharp vertices. Examples include cubes, prisms, and pyramids. However, shapes such as cones and spheres do not qualify as polyhedrons because they lack polygonal faces. It can have any polygon such as a triangle, pentagon, hexagon, etc. as faces as well and it satisfies Euler’s formula, which will be discussed later in the article.
In mathematics, polyhedrons have received a great deal of attention and are used in various fields such as Physics, computer graphics, crystallography, architecture, and other disciplines. In this article, we will discuss all the concepts related to polyhedrons including polyhedron definition, polyhedron shape, types of polyhedrons, their faces, edges, vertices, and real-life examples of polyhedrons.
Table of Content
What is a Polyhedron?
A polyhedron is a three-dimensional solid consisting of a group of polygons, made up of vertices at the intersections of the flat faces and straight edges.
Due to their unique features and potential uses, polyhedrons have been investigated and explored in several disciplines, including mathematics, architecture, computer graphics, and engineering.
Polyhedron Meaning
The term “polyhedron” precisely describes these geometric objects’ distinctive feature, which is the collection of numerous flat faces.
A triangle, square, pentagon, or any other regular or irregular polygon is a type of polygon that characteristics each face. The edges that connect these faces come together at vertices or corners.
Polyhedron Shape
Polyhedrons can be found in many different kinds of complex shapes. The Platonic solids (cube, octahedron, dodecahedron, and icosahedron) are regular polyhedrons having symmetrical vertices, edges, and faces as well as identical regular polygonal faces. The faces and edges of a polyhedron are mostly asymmetrical; they are not necessarily congruent or symmetric.
Examples: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron, Prism, etc.
Polyhedron Examples
There are various examples of polyhedrons, some of the most common examples are listed in the following table:
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Polyhedrons |
Characteristics |
Shape or Form |
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Tetrahedron |
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Octahedron |
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Dodecahedron |
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Icosahedron |
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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 make and has 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.
Detailed examples of platonic solids are discussed under “Examples of Polyhedron”.
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Types of Polyhedron
Polyhedrons can be classified into various categories, based on various parameters.
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Based on Edge Length
- Regular Polyhedron
- Irregular Polyhedron
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Based on 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 had 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 true:
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.
Practice Problems on Polyhedrons
Problem 1: What is a polyhedron, and how does it differ from other three-dimensional shapes?
Problem 2: Name and describe the characteristics of the five platonic solids.
Problem 3: If a polyhedron has 12 edges and 6 vertices, how many faces does it have?
Problem 4: Can a polyhedron have exactly 7 faces? Explain your reasoning.
Problem 5: What is the Euler’s formula for polyhedra, and how is it used to relate the number of faces, edges, and vertices?
Summary – Polyhedron
Polyhedrons are three-dimensional geometric figures defined by flat polygonal faces, straight edges, and distinct vertices. They come in various forms, including regular, irregular, convex, and concave configurations, influencing their structural properties and symmetry. Common types of polyhedrons include cubes, prisms, and pyramids, each distinguished by the shape and arrangement of their faces. These shapes are fundamental in various fields such as architecture, art, and mathematics, particularly in studying spatial relationships and building structures. Unlike polyhedrons, other geometric forms like cones and spheres are not considered polyhedrons because they lack the necessary polygonal faces.
FAQs on Polyhedron – Definition, Shapes, Formula and Examples
What is the definition of polyhedron?
The term “polyhedron” precisely describes these geometric objects’ distinctive feature, which is the collection of numerous flat faces. A triangle, square, pentagon, or any other regular or irregular polygon is a type of polygon that characteristics each face. The edges that connect these faces come together at vertices or corners.
What is the Relationship between a Polyhedron and a Polygon?
Each face of a polyhedron is made up of a polygon, and these polygons are joined together by edges to form the polyhedron’s overall shape.
Is a Pyramid a Polyhedron?
Yes, a pyramid is a polyhedron. A pyramid is a three-dimensional shape with flat polygonal faces, straight edges, and vertices, hence it fits the definition of a polyhedron.
What are the Main types of Polyhedrons?
Regular Polyhedrons, Convex and Concave Polyhedrons, Prisms, Pyramids, Archimedean solids, Johnson solids, and stellated polyhedrons are the main varieties of polyhedrons.
Can a Polyhedron have 10 Faces?
Yes, a polyhedron can have 10 faces.
Example: Pentagonal Prism
Is Cylinder a Polyhedron?
No, Cylinder is not a polyhedron because it is curved and polyhedrons have flat surfaces.