# Year 10+ 3D Geometry

### Chapters

### Euler's Formula

# Euler's Formula

`Euler's Formula`

is a relationship between the numbers of faces, edges and vertices (corners) of a **convex** polyhedron
(a 3-D shape with flat faces and straight edges that doesn't have any dents in it).

It states that the **number of vertices** plus the **number of faces** minus the **number of edges** always equals
\(2\).

We write Euler's Formula as

A **tetrahedron** is a convex polyhedron. It has \(4\) vertices, \(4\) faces and \(6\) edges. Let's test out Euler's
Formula:

## The Platonic Solids

Now, let's try Euler's formula on the rest of the Platonic solids.

### The Cube (or Hexahedron)

The cube has \(8\) vertices, \(6\) faces and \(12\) edges. So,

### The Octahedron

The octahedron has \(6\) vertices, \(8\) faces and \(12\) edges. So,

### The Dodecahedron

The dodecahedron has \(20\) vertices, \(12\) faces and \(30\) edges. So,

### The Icosahedron

The icosahedron has \(12\) vertices, \(20\) faces and \(30\) edges. So,

## Are There Solids that Euler's Formula Doesn't Work For?

Yes, Euler's formula is only guaranteed to work for convex polyhedrons. Here's a polyhedron that isn't convex. It's called a
**small stellated dodecahedron**:

The small stellated dodecahedron (try saying that 10 times quickly) has 12 vertices, 12 faces and 30 edges. So,

**lots**of dents.

## Euler Characteristic

Mathematicians love to generalise things. One generalisation of Euler's formula is the `Euler Characteristic`

. The Euler characteristic
can take on all sorts of different values (not just 2) and we can find the Euler characteristic of non-convex polyhedra and other solids that are not
polyhedra. The Euler characteristic is defined to be

Mathematicians are interested in the Euler characteristic as it doesn't change if you deform a solid: twist or bend it, squeeze it or shrink it. It is
what we call a **topological invariant.** Let's have a look at a few examples of Euler characteristics

Solid | Picture | \(\chi\) |
---|---|---|

Sphere | \(2\) | |

Klein Bottle | \(0\) | |

3-holed Torus | \(-4\) | |

Small Stellated Dodecahedron | \(-6\) |

**Note:** we know that the Euler characteristic of a sphere is \(2\) because we can pump up any of the Platonic solids to form a sphere,
and their Euler characteristics are all \(2\). We say that the Platonic solids are `homeomorphic`

to a sphere because we can deform them (by pumping
them up) to form a sphere.

### Description

There are several lessons related to 3D geometry such as

- Euler's formula
- Vertices, Edges and Faces
- Volumes of 3D shapes
- etc

Even though we've titled this lesson series to be more inclined for Year 10 or higher students, however, these lessons can be read and utilized by lower grades students.

### Prerequisites

Understanding of 3D shapes

### Audience

Year 10 or higher, but suitable for Year 8+ students

### Learning Objectives

Get to know 3D Geometry

Author: Subject Coach

Added on: 27th Sep 2018

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