# Polyhedron

‘Poly’ – many, and ‘hedron’ – base or seat. So, a polyhedron is a 3-dimensional solid with multiple flat faces.

## Definition

A polyhedron (plural – 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 – 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 – It is a polyhedron with faces that are regular polygons congruent to each other. Regular polyhedrons are also known as ‘platonic solids’. Cubes, tetrahedrons, and octahedrons are common examples of regular polyhedrons.

2. Irregular Polyhedron – 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 – 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 – 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
∴ V = 2 + E – F
= 2 + 30 – 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
∴ V = 2 + E – F
= 2 + 30 – 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.