# Polyhedron

â€˜Polyâ€™ – many, and â€˜hedronâ€™ – base or seat. So, a polyhedron is a 3-dimensional solid with multiple flat faces.

## Definition

A polyhedron (plural â€“ polyhedra or polyhedrons) is a 3-dimensional shape consisting of polygons joined at their edges. A polyhedron has no curved face. Its components are the multiple polygon-shaped flat faces.

## Parts

It has 3 parts â€“ face, edge, and vertex.

1. Face – The flat surface of a polyhedron.
2. Edge – The region where 2 faces meet.
3. Vertex (Plural – vertices).- The point of intersection of 2 or more edges. It is also known as the corner of a polyhedron.

Polyhedrons are named based on the number of faces they have, such as Tetrahedron (4 faces), Pentahedron (5 faces), and Hexahedron (6 faces).

Platonic solids, prisms, and pyramids are 3 common groups of polyhedrons.

## Types

### Regular and Irregular

1. Regular Polyhedron â€“ It is a polyhedron with faces that are regular polygons congruent to each other. Regular polyhedrons are also known as â€˜platonic solidsâ€™. Cubes, tetrahedrons, and octahedrons are common examples of regular polyhedrons.

2. Irregular Polyhedron â€“ It is a polyhedron with faces that are irregular polygons where all the components are not the same. Prisms and pyramids are common examples of irregular polyhedrons.

### Convex and Concave

1. Convex Polyhedrons â€“ It is a polyhedron where a line segment joining any 2 points on its surface lies completely inside it. All regular polyhedra are convex.

2. Concave Polyhedron â€“ It is a polyhedron where a line segment joining any 2 points on its surface can lie outside it.

## Formulas

Like all other 3-dimensional shapes, we can calculate the surface areas and volumes of polyhedrons, such as a prism and a pyramid, using their specific formulas.

### Euler’s Polyhedron Formula

We can calculate the number of faces, edges, and vertices of any polyhedron using the formula based on Euler’s theorem:

A 20-sided polyhedron has 30 edges. How many vertices does it have?

Solution:

As we know that the given polyhedron is twenty sided, so it will have 20 faces,
V + F – E = 2, here F = 20, E = 30
âˆ´ V = 2 + E â€“ F
= 2 + 30 â€“ 20
= 12 vertices

How many vertices does the polyhedron have if it is a 12-sided polyhedron with 30 edges?

Solution:

As we know that the given polyhedron is a dodecahedron,
V + F – E = 2, here F = 12, E = 30
âˆ´ V = 2 + E â€“ F
= 2 + 30 â€“ 12
= 20 vertices

## Real-life Examples

• Soccer balls
• The Great Pyramid of Giza
• Prisms used in Physics laboratory
• Polyhedron-shaped buildings
• Goldberg polyhedron

## FAQs

Q1. Is a cone a polyhedron?

Ans. No. A cone is not a polyhedron because it has a curved face and a circular base.

Q2. Is the cylinder a polyhedron?

Ans. No. A cylinder is not a polyhedron because it has a curved face and 2 circular bases.

Q3. Is a rectangular pyramid a polyhedron?

Ans. Yes. A rectangular pyramid has 5 flat faces. So it is a pentahedron.