 # A Naturally Occurring Function

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A Naturally Occurring Function
Euler’s Formula A Naturally Occurring Function

Leonhard Euler was a brilliant Swiss mathematician
Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Euler discovered an interesting relationship between the number of faces, vertices, and edges for any polyhedron.

Poly-what?

A polyhedron is a 3 dimensional shape with flat sides.
A polyhedron is a 3 dimensional shape with flat sides.

All prisms and pyramids are examples of polyhedra (plural for polyhedron).
PRISMS PYRAMIDS

Any polyhedron has faces, vertices, and edges.
VERTEX

A face is a flat side.

This rectangular prism has 6 faces.

This rectangular prism has 6 faces.
FRONT

This rectangular prism has 6 faces.
BACK FRONT

This rectangular prism has 6 faces.
TOP BACK FRONT

This rectangular prism has 6 faces.
TOP BACK FRONT BOTTOM

This rectangular prism has 6 faces.
TOP BACK FRONT LEFT BOTTOM

This rectangular prism has 6 faces.
TOP BACK FRONT RIGHT LEFT BOTTOM

This rectangular prism has 6 faces.
TOP BACK FRONT RIGHT LEFT BOTTOM

This rectangular prism has 6 faces.
TOP BACK FRONT RIGHT LEFT BOTTOM

This rectangular prism has 6 faces.
TOP LEFT BACK FRONT RIGHT BOTTOM

This rectangular prism has 6 faces.
TOP LEFT BACK FRONT RIGHT BOTTOM

This rectangular prism has 6 faces.
TOP RIGHT LEFT BACK FRONT BOTTOM

This square pyramid has 5 faces.
The faces consist of 4 triangles and a square.

The faces consist of 4 triangles and a square.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

A triangular pyramid has 4 faces.

An edge is a line segment where two faces meet.

A rectangular prism has 12 edges.

A triangular pyramid has 6 edges.

A vertex is a corner. It is a point that connects 2 or more edges.

A vertex is a fancy word for “corner.”
B Every triangle has 3 vertices (corners). Points A, B, and C are vertices.

A rectangular prism has 8 vertices.

A rectangular prism has 8 vertices.

A triangular pyramid has 4 vertices.

A triangular pyramid has 4 vertices.

Euler studied the faces, vertices, and edges of different polyhedra.

Like most great mathematicians and scientists, he organized his data in a chart.
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9

Euler looked for a relationship between these numbers.
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9

Can you determine Euler’s formula that relates the # of Faces and # of Vertices to the # of Edges?
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9

Faces + Vertices –2 = Edges
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 - 2 = + - 2 = + - 2 = +

Use Euler’s Formula to determine the number of edges in a pentagonal prism.
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 Pent. Prism 7 10 - 2 = + - 2 = + - 2 = +

Use Euler’s Formula to determine the number of edges in a pentagonal prism.
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 Pent. Prism 7 10 - 2 = + - 2 = + - 2 = + - 2 = +

Use Euler’s Formula to determine the number of edges in a pentagonal prism.
Polyhedron # of Faces # of Vertices # of Edges Cube 6 8 12 Sq. Pyramid 5 Tri. Prism 9 Pent. Prism 7 10 15 - 2 = + - 2 = + - 2 = + - 2 = +

Euler’s Formula works for any polyhedron.
SUMMARY: Euler’s Formula says that if you add the number of faces and vertices, then subtract by 2, the result is the number of edges. Euler’s Formula works for any polyhedron.

THE END!