Polyhedron

Polyhedron definition states that “a three-dimensional structure in Euclidean geometry, made up of a finite number of polygonal faces”.

The boundary between the interior and the exterior of a solid is a polyhedron.

Polyhedrons, in general, are named according to the number of faces.

Parts of Polyhedron

The Polyhedron has three parts namely:

• Face

The face is a flat surface that makes up a polyhedron which is regular polygons.

• Edge

Edge is the region where the two flat surfaces meet to form a line segment.

• Vertex

Vertex, also known as a corner, is a point of intersection of the edges of the polyhedron.

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Types of Polyhedron

There are three main types of Polyhedron:

1. Prisms

A prism is a polyhedron consisting of an n-sided polygonal base, a second base that is a translated copy of the first, and no other faces that connect the two bases to corresponding sides.

Prisms are named according to their cross-sections(polyhedron faces).

• A prism with a square cross-section is called a Square prism(cuboid).

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• A prism with a square cross-section(along the length too) is called a cube.

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• A prism with a triangular section is called a Triangular prism.

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• A prism with a Pentagonal section is called a Pentagonal prism.

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2. Platonic Solids

A regular, convex polyhedron is a Platonic solid in three-dimensional space. It is constructed of congruent, regular, polygonal faces that meet at each vertex with the same number of faces.

Platonic solids are of five types based on Polyhedron faces and polyhedron shapes:

• Tetrahedron

• It has 4 faces, 4 Vertices, and 6 Edges.

• It has 3 triangles that meet at each vertex.

• The surface area of tetrahedron= \[\sqrt{3} a^{2}\]

• The volume of tetrahedron= \[\frac{\sqrt{2}}{12} a^{3}\]

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• Cube

• It has 6 faces, 8 Vertices, 12 Edges.

• It has three squares that meet at each vertex.

• The surface area of the cube= 6a2

• The volume of the cube = a3

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• Octahedron

• It has 8 faces, 6 Vertices, 12 Edges.

• It has 4 triangles that meet at each vertex.

• The surface area of Octahedron = \[2 sqrt{3} a^{2}\]

• The volume of Octahedron = \[\frac{\sqrt{2}}{3} a^{3}\]

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• Dodecahedron

• It has 12 faces, 20 Vertices, 30 Edges.

• It has 3 pentagons that meet at each vertex.

• The surface area of Dodecahedron= \[3 \sqrt{25 + 10\sqrt{5}} a^{2}\]

• The volume of Dodecahedron = \[\frac{15 + 7\sqrt{5}}{4} a^{3}\]

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• Icosahedron

• It has 20 faces, 12 Vertices, 30 Edges.

• It has 5 triangles that meet at each vertex.

• The surface area of Icosahedron = \[5 \sqrt{3} a^{2}\]

• The volume of Icosahedron = \[\frac{5(3 + \sqrt{5})}{12} a^{3}\]

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3. Pyramids

A pyramid is a polyhedron created by connecting a polygonal base and a point, called the apex. A triangle, called a lateral face, is formed by any base edge and apex. It is a conical solid with a foundation of polygons.

The types of pyramids are named after the base of a pyramid.

• Triangular Pyramid

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• Square Pyramid

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• Pentagonal Pyramid

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• Right Pyramid

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• Oblique Pyramid

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Counting Polyhedron Faces, Edges, and Vertices

Euler’s formula relates the number of faces, vertices, and edges of any polyhedron. This formula is used in Counting Polyhedron Faces, Edges, and Vertices.

Euler’s formula is given as follows: 

F + V - E = 2

Where F = Number of Faces

V = Number of Vertices

E = Number of Edges

Problems on Polyhedron Faces, Edges, and Vertices

1) The Polyhedron has 6 faces and 12 edges. Find the number of Vertices. Also, name the type of Polyhedron.

Ans: Here to find the number of vertices we will use Euler’s formula,

F+ V - E = 2

From the question F = 6, E = 12, V = ?. Substituting these values in the Euler’s formula we get,

6 + V - 12 = 2

V - 6 = 2

V = 8

Here we can conclude that the Polyhedron is a Cube.

2) The Polyhedron has 5 faces and 6 vertices. Find the number of edges. Also, name the type of Polyhedron.

Ans: Here we will use Euler’s formula to find the number of edges,

F + V - E = 2

From the given data F = 5, V = 6, E = ?. Substituting these values in the Euler’s formula we get,

5 + 6 - E = 2

11 - E = 2

E = 9

Therefore the polyhedron is a Triangular Prism.

Fun Facts

• The word Polyhedron comes from the Greek words “poly” meaning many, and “hedron,” meaning surface.

• Polyhedron means that to form a 3-dimensional shape, numerous flat surfaces are connected.

• Polyhedra is the plural of a polyhedron.