Polyhedron – Definition, Examples, Practice Problems, FAQs

What is a Polyhedron?

We know that a polygon is a flat, plane, two-dimensional closed shape bounded by line segments. Common examples of polygons are square, triangle, pentagon, etc. 

Now, can you imagine a three dimensional figure with faces in the shape of a polygon? Such a three-dimensional figure is known as a Polyhedron. Examples you must be familiar with are diamonds, rubik’s cube, pyramids, etc.

Polyhedron Definition

A three-dimensional shape with flat polygonal faces, straight edges, and sharp corners or vertices is called a polyhedron. 

Common examples are cubes, prisms, pyramids. However, cones, and spheres are not polyhedrons since they do not have polygonal faces.

The plural of a polyhedron is called polyhedra or polyhedrons. 

Faces, Edges, and Vertices

The parts of a polyhedron are classified as faces, edges, and vertices.

• Polyhedron Faces: The flat surfaces of a polyhedron are termed as its faces, which are basically polygons.
• Polyhedron Edges: Edges are the line segments where two faces meet. 
• Polyhedron Vertices: The point of intersection of two edges is known as a vertex.

The figure given below will give us a better view:

Prisms, Pyramids, and Platonic Solids

Polyhedrons can be broadly classified as prisms, pyramids, and Platonic solids. Let’s understand each type.

1. Prism

A prism has its both ends (base and top) as identical polygons and its side faces are flat (rectangles or parallelograms). Prisms are named after its base, which can be a triangle, square, rectangle, or any n-sided polygon.

2. Pyramid

A pyramid has its base as any polygon and its side faces are triangles with a common vertex (known as apex). If the base of a pyramid is an n-sided polygon, then it has (n+1) faces, (n+1) vertices, and 2n edges.

3. Platonic Solids

In Platonic solids, all the faces are congruent regular polygons and the same number of faces meet at each vertex. 

Examples in Real Life

We can observe several polyhedra in our daily existence such as a Rubik’s cube, dice, buckyball, pyramids, and so on. A diamond is one of the real-life examples of polyhedron. 

Types of Polyhedron

Polyhedra are mainly divided into two types based on the polygonal faces and the base ​​— regular polyhedrons and irregular polyhedrons. Platonic solids are regular polygons. Prisms and pyramids are irregular polyhedrons.

Regular Polyhedron

Regular polyhedrons are made up of regular polygons. They are also known as “Platonic solids.” They have all their faces, edges, and angles congruent. The following is the list of the five regular polyhedrons:

Irregular Polyhedron

An irregular polyhedron has polygonal faces that are not congruent to each other. It is made up of polygons having different shapes. So, all its components are not the same.

Polyhedrons can also be divided into convex and concave categories, just like polygons. 

Convex Polyhedron

A convex polyhedron is similar to a convex polygon. If a line segment that joins any two points on the surface lies inside the polyhedron, then it is known as a convex polyhedron. All platonic solids are convex.

Concave Polyhedron

A concave polyhedron is similar to a concave polygon. If a line segment that joins any two points on the surface lies outside the polyhedron, it is known as a concave polyhedron. 

Euler’s Formula

There is a relationship between the number of faces, edges, and vertices in a polyhedron, which can be presented by a math formula known as “Euler’s Formula.”

$\text{F} + \text{V}$ $–$ $\text{E} = 2$ 

where, 

$\text{F} =$ number of faces 

$\text{V} =$ number of vertices 

$\text{E} =$ number of edges 

If we know any two values among F, V, or E, we can find the third missing value using Euler’s formula. So, the questions like ‘How many edges does a polyhedron have?’ or ‘How many faces does a polyhedron have?’ can be answered easily if the other two values are known. We can also check if a polyhedron with the given number of parts exists or not. 

For example, a cube has 8 vertices, 6 faces, and 12 edges. 

$\text{F} = 6, \text{V} = 8, \text{E} = 12$

Applying Euler’s formula, we get $\text{F} + \text{V}$ $–$ $\text{E} = 2$ 

Substituting the values in the formula: $6 + 8$ $–$ $12 = 2 \Rightarrow 2 = 2$. 

Hence, the cube is a polyhedron. 

Solved Examples on Polyhedron

1. How many types of regular polygons are there?

Solution: There are 5 types of regular polygons: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 

2. Check if the polyhedron with 10 vertices, 8 edges, and 4 faces exists or not.

Solution: We will apply Euler’s formula: 

$\text{V} = 10, \text{E} = 8$, and $\text{F} = 4$

$\text{F} + \text{V}$ $–$ $\text{E}  = 4 + 10$ $–$ $8 = 6 ≠ 2$.

The polyhedron with the given dimensions does not exist. 

3. How many faces does a polyhedron with 12 vertices and 18 edges have?

Solution: $\text{V} = 12$, $\text{E} = 18$, and $\text{F} =$ ?

By applying Euler’s formula, we get

 $\text{F} + \text{V}$ $–$ $\text{E} = 2$ 

$\Rightarrow\text{F} + 12$ $–$ $18 = 2$ 

$\Rightarrow\text{F} = 8$

Number of faces = 8

Practice Problems on Polyhedron

Frequently Asked Questions on Polyhedron