# Polyhedrons

A **polyhedron** is a solid with flat faces

(from Greek poly- meaning "many" and -hedron meaning "face").

Each face is a polygon (a flat shape with straight sides).

## Examples of Polyhedra:

Its faces are all squares

Its faces are triangles

and rectangles

What faces does it have?

So **no curved surfaces**: cones, spheres and cylinders are **not** polyhedrons.

## Common Polyhedra

Platonic Solids | ||

Prisms | ||

Pyramids |

Note: the plural of **polyhedron** is either **polyhedrons** or **polyhedra**

## Many More

Explore 100s of Animated Polyhedron Models. You can also see some Images of Polyhedra if you want. |

## Counting Faces, Vertices and Edges

When we count the number of faces (the flat surfaces), vertices (corner points), and edges of a polyhedron we discover an interesting thing:

The number of **faces**

*plus* the number of **vertices**

*minus* the number of **edges** equals **2**

This can be written neatly as a little equation:

### F + V − E = 2

It is known as Euler's Formula (or the "Polyhedral Formula") and is very useful to make sure we have counted correctly!

### Example: Cube

A cube has:

- 6 Faces
- 8 Vertices (corner points)
- 12 Edges

F + V − E = 6 + 8 − 12 = **2**

### Example: Triangular Prism

This prism has:

- 5 Faces
- 6 Vertices (corner points)
- 9 Edges

F + V − E = 5 + 6 − 9 = **2**

But there are cases where it does **not** work! Read Euler's Formula for more.

## Diagonals

A diagonal is a straight line inside a shape that goes from one corner to another (but not an edge).

A polyhedron can have lots of diagonals. Can you think of one without diagonals?