# Polyhedron

A **polyhedron** , in the area of Euclidean geometry, is defined as a **three-dimensional** object that is composed of a finite number of **polygonal** surfaces or, in other words, **faces** . Technically, a polyhedron is the boundary between the **inside** and the outside of a solid. In general, polyhedra are named according to the number of faces they have. A tetrahedron has **four faces** , a pentahedron five, and so on; A cube is a six-sided regular polyhedron ( **hexahedron** ) whose faces are **square** . Faces meet on line segments called **edges** , which meet at points called**vertices** .

## What is a polyhedron?

A polyhedron is an element within the area of **geometry** that has different number of flat **faces** and that have the ability to store a large amount of volume that is not **infinite** .

- Definition
- Elements of a polyhedron
- Types of polyhedron
- features
- Area of a polyhedron
- Examples

## Definition

In mathematics , we define a polyhedron as a **solid** that has a series of **plane faces** . When we refer to the term solid we are referring to a **geometric** shape that is **three-dimensional** . When a shape is three-dimensional, it means that it has space within it. It is not a flat object that is simply drawn on a flat sheet of paper. In other words, three-dimensional solids are things you can hold.

We can say that it is a solid that has **flat faces** . The word comes from the Greek, **poly** which means ” **many** ” and – **edro** which means *” face “* . For a body to be a polyhedron there does not have to be any

**curved**

**surface**in it.

## Elements of a polyhedron

In a polyhedron we can find the following elements:

**Faces**: are the**polygons**that are forming the polyhedron.**Edges**: are the segments in which the faces of the polyhedron**intersect**.**Vertices**: are the points where the edges of the polyhedron**intersect**.**Dihedral angle**: this is the angle formed by**two faces**that intersect. There are as many dihedral angles as the number of edges.**Polyhedron angle**: are the angles determined by the**faces**that**affect the**same vertex. There are as many as the number of vertices.

## Types of polyhedron

The types of polyhedron that exist are the following:

#### Regular polyhedron

A regular polyhedron is one whose faces are all **regular** polygons and they are all the same. The **edges** are also all the same and there are only five types of regular polyhedra:

- Regular
**tetrahedron**: it is a regular polyhedron whose surface is formed by four equilateral triangles of the same size. **Cube**(or regular hexahedron): it is made up of six equal squares.**Regular octahedron**: the surface is made up of eight equal equilateral triangles**Regular dodecahedron**: it is formed by twelve equal regular pentagons**Regular cosahedron**: it is a regular polyhedron in which the faces are twenty equilateral triangles, all of them equal.

#### Irregular polyhedron

Irregular ones are those whose faces are polygons but they are not all the **same** . Among them we can mention the Archimedean solids that are convex polyhedra with regular but not uniform faces. They are classified mainly by the number of faces that their surface has:

**Tetrahedron**: irregular polyhedron with four faces**Pentahedron**: irregular with five faces**Hexahedron**: irregular with six faces**Heptahedron**: irregular with seven faces**Octahedron**: irregular polyhedron with eight faces**Eneahedron**: irregular with nine faces**Decahedron**: irregular with ten faces

In addition, there are two very special cases of irregular **tetrahedron** :

**Trirectangle tetrahedron: it**is the one that has three faces that are**right triangles**, in which their right angles concur to the same vertex.**Isofacial**tetrahedron: it is an irregular tetrahedron whose base is a**right triangle**and its three**lateral**faces are three equal isosceles triangles.**Convex**: they are formed when any pair of points in**space**that are inside the body are joined by an internal**line**segment .**Polyhedron with regular faces:**when all the faces of the same are regular polygons.**Polyhedron of uniform faces:**when all the faces are equal.

## features

The main characteristics of the polyhedron are the following:

- Their faces are
**flat**. - It has a finite
**volume**of flat surfaces. - They are
**three-dimensional**bodies . - They are named depending on the number of faces they have: tetrahedron, pentahedron, hexahedron, heptahedron, icosahedron.

## Area of a polyhedron

The calculation of areas of polyhedra does not need any new **formula** since it is enough to calculate the **areas** of all the faces and add. This is done using the area formulas of polygons.

What is the area of the cube with edges of measure 5 cm? It is a regular polyhedron, with 6 faces. Each face is a square and therefore the area is 52 = 25cm. We have 6 faces therefore = 6 * 25 = 150 cm ^{2}

What is the area of the octahedron with edges of measure 5 cm? It is regular, with 8 faces. As seen in part, the octahedron consists of 8 triangles. Simply calculate the area of each triangle and multiply by 8.

## Examples

Among the most common examples of irregular polyhedra we can mention the Platonic solids, the **blunt cube** , **truncated icosahedron** , **prisms** and **anti-prisms** .