Geometry

Hong Kong

Stage 1 - Stage 3

Lesson

So far we have been able to name solids and describe them as being right or oblique, concave or convex and whether or not they are platonic. There are some more key terms we need to know about with relation to solids.

A vertex is a point where two or more straight lines meet. Otherwise known as a corner.

This rectangular based pyramid has $5$5 vertices.

An edge is a line segment that joins two vertices.

This rectangular based pyramid has $8$8 edges.

A face is any of the individual surfaces of a solid object. This rectangular based pyramid has $5$5 faces.

A polyhedron is the 3D equivalent of a polygon. Remember how a polygon is a many angled (many straight sided) figure. Well a polyhedron is a many sided flat faced figure. A polyhedron has no curved edges and each face is a polygon.

Here are some examples of 3D solids that **are **polyhedra:

These ones are **not**. These all have curved faces or edges.

As we just discovered in the investigation, there is a special relationship between the number of vertices, edges and faces together.

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as calculus and graph theory. This relationship between vertices, edges and faces is name after him.

$V-E+F=2$`V`−`E`+`F`=2

Sometimes this formula is easier to remember in an alternate form, it is exactly the same though, you can use which ever rule suits the purpose or question.

$V+F=E+2$`V`+`F`=`E`+2

Even though this formula looks very simple, it summarises a property of polyhedra that kept mathematicians intrigued and entertained for thousands of years.

- $V$
`V`is the number of vertices - $E$
`E`is the number of edges - $F$
`F`is the number of faces

Euler's rules holds true for all simple polyhedra.

(Simple polyhedra are polyhedra that do not have holes running through them.)

For any rectangular based pyramid, show that $V+F=E+2$`V`+`F`=`E`+2

a) first find the number of vertices

**Think**: Vertices are the points.

**Do**: There are $5$5 vertices in total

b) Find the number of faces

**Think**: There are $4$4 triangular faces and $1$1 base

**Do**: There are $5$5 faces in total

c) Find the number of edges,

**Think**: There are $4$4 sloping edges that connect to the vertex, and $4$4 edges around the base.

**Do**: There are $8$8 edges in total

d) Confirm that $V+F=E+2$`V`+`F`=`E`+2

$V+F$V+F |
$=$= | $5+5$5+5 |

$=$= | $10$10 | |

$E+2$E+2 |
$=$= | $8+2$8+2 |

$=$= | $10$10 | |

Therefore | ||

$V+F$V+F |
$=$= | $E+2$E+2 |

Therefore Euler's Theorem holds true!