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A polyhedron is a 3D shape that has flat faces, straight edges, and sharp vertices (corners). The word "polyhedron" is derived from a Greek word, where 'poly' means "many" and hedron means "surface".Thus, when many flat surfaces are joined together they form a polyhedron. These shapes have names according to their faces that are usually polygons. The most common names are cubes, hexahedrons, etc. Let us learn more about the types of polyhedrons and solve a few examples to understand the shape better.
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A polyhedron is a three-dimensional solid made up of polygons. It has flat faces, straight edges, and vertices. For example, a cube, prism, or pyramid are polyhedrons. Cones, spheres, and cylinders are non-polyhedrons because their sides are not polygons and they have curved surfaces. The plural of a polyhedron is also known as polyhedra. They are classified as prisms, pyramids, and platonic solids. For example, triangular prism, square prism, rectangular pyramid, square pyramid, and cube (platonic solid) are polyhedrons. Observe the following figure which shows the different kinds of polyhedrons.
Parts of a Polyhedron
The dimensions of a polyhedron are classified as faces, edges, and vertices.
- Face: The flat surface of a polyhedron is termed as its face.
- Edge: The two faces meet at a line called the edge.
- Vertices: The point of intersection of two edges is a vertex.
Observe the following figure which shows the face, vertex, and edges of a shape.
Types of Polyhedron
Polyhedra are mainly divided into two types – regular polyhedron and irregular polyhedron.
A regular polyhedron is also called a platonic solid whose faces are regular polygons and are congruent to each other. In a regular polyhedron, all the polyhedral angles are equal. There are five regular polyhedrons. The following is the list of five regular polyhedrons.
- Tetrahedron: A tetrahedron has 4 faces, 6 edges, and 4 vertices (corners); and the shape of each face is an equilateral triangle.
- Cube: A cube has 6 faces, 12 edges, and 8 vertices; and the shape of each face is a square.
- Regular Octahedron: A regular octahedron has 8 faces, 12 edges, and 6 vertices; and the shape of each face is an equilateral triangle.
- Regular Dodecahedron: A regular dodecahedron has 12 faces, 30 edges, 20 vertices; and the shape of each face is a regular pentagon.
- Regular Icosahedron: A Regular icosahedron has 20 faces, 30 edges, and 12 vertices; and the shape of each face is an equilateral triangle.
Observe the following figure which shows the various types of regular polyhedrons.
A polyhedron with irregular polygonal faces that are not congruent to each other, and in which the polyhedral angles are not equal is called an irregular polyhedron.
A convex polyhedron is just like a convex polygon. If a line segment joining any two points on the surface of a polyhedron entirely lies inside the polyhedron, it is called a convex polyhedron.
A concave polyhedron is quite similar to a concave polygon. If a line segment joining any two points on the surface of a polyhedron goes outside the polyhedron, it is called a concave polyhedron.
There is a relationship between the number of faces, edges, and vertices in a polyhedron. We can represent this relationship as a math 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 using the Euler's Formula we can easily find the missing part of a polyhedron. We can also verify if a polyhedron with the given number of parts exists or not. For example, a cube has 6 faces, 8 vertices (corner points) and 12 edges. Let us check whether a cube is a polyhedron or not by using 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
Substituting the values in the formula: 6 + 8 - 12 = 2 ⇒ 2 = 2. Hence proved, cube is a polyhedron.
Check out the following articles related to the Polyhedron.
Example 1: An eight-faced polyhedron has 12 edges. How many vertices does it have?
Solution: Given, number of faces (F) = 8; edges (E) = 12 and vertices (V) = ?
Let us apply the Euler's formula.
F + V - E = 2
8 + V - 12 = 2
V - 4 = 2
V = 2 + 4
V = 6
Therefore, the polyhedron has 6 vertices.
Example 2: The number of dimensions of a polyhedron are given as follows: edges (E) = 4, faces (F) = 6, and vertices (V) = 8. Check and tell if a polyhedron with these dimensions exists?
Solution: We can use the Euler's formula and apply these values: E = 4, F = 6, and V = 8
F + V - E = 2
6 + 8 - 4 = 14 - 4 = 10
These values do not satisfy the Euler's formula, therefore, a polyhedron with the above number of dimensions does not exist.
Example 3: A polyhedron has 14 vertices and 20 edges. How many faces does it have?
Solution: We can use the Euler's formula to find the faces.
F + V - E = 2
Where F = Faces, V = Vertices and E = Edges
Given, V = 14 and E = 20
F + 14 - 20 = 2
F - 6 = 2
F = 6 + 2 = 8
Therefore, the polyhedron has 8 faces.
Practice Questions on Polyhedron
FAQs on Polyhedron
What is a Polyhedron?
A polyhedron is a 3D-shape consisting of flat faces shaped as polygons, straight edges, and sharp corners or vertices. A shape is named a polyhedron according to the number of faces it has. Ideally, this shape is the boundary between the interior and exterior of a solid.
How Do You Identify a Polyhedron?
Any 3D shape equivalent to a polygon with straight sides is considered a polyhedron. The most common example is a cube that consists of straight edges and flat faces. Three-dimensional shapes that have curved faces do not come under polyhedrons. For example, cone and cylinder do not come under the category of polyhedrons.
How Many Faces Does a Polyhedron Have?
Depending on the number of flat sides of the polygonal-based polyhedron, we can count the number of faces. We can also find the number of faces in a polyhedron using Euler's formula, F + V - E = 2, if we know the number of edges and vertices.
How Many Regular Polyhedrons are There?
There are 5 regular polyhedrons, they are: Tetrahedron (or pyramid), Cube, Octahedron, Dodecahedron, and Icosahedron.
Is Sphere a Polyhedron?
No, a sphere is not a polyhedron because it has a curved surface, whereas, polyhedrons have only straight and flat surfaces, for example, a cube is a polyhedron.
Is Prism a Polyhedron?
Yes, a prism is a polyhedron. A prism has flat faces, straight edges, and sharp vertices. Shapes like a triangular prism, rectangular prism, and pentagonal prism come under polyhedrons.
Is a Polyhedron a Pyramid?
Yes, a pyramid is a polyhedron because the base of the pyramid is a polygon and all lateral faces are triangles.
Is a Cylinder a Polyhedron?
A cylinder is not a polyhedron because it has a curved surface. Polyhedrons have flat surfaces, called faces, which are made of polygons. For example, prisms and pyramids come under polyhedrons.