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Euler’s Formula A Naturally Occurring Function

Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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Page 1: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Euler’s Formula

A Naturally Occurring Function

Page 2: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Leonhard Euler was a brilliant Swiss mathematician. He is

often referred to as the “Beethoven of Mathematics.”

         

http://encarta.msn.com/index/conciseindex/11/MediaMax.asp?pg=3&ti=761556057&idx=461515527
Page 3: Euler’s Formula A Naturally Occurring Function. 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.

Page 4: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Poly-what?

Page 5: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A polyhedron is a 3 dimensional shape with flat sides.                 

http://www.hopepaul.com/kids/solidgeo/solidgeo.htm
http://www.hopepaul.com/kids/solidgeo/solidgeo.htm
Page 6: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

All prisms and pyramids are examples of polyhedra (plural for

polyhedron).

POLYHEDRA

PRISMS PYRAMIDS

Page 7: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Any polyhedron has faces, vertices, and edges.

EDGE

FACE

VERTEX

Page 8: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A face is a flat side.

Page 9: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

Page 10: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

FRONT

Page 11: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

FRONT

BACK

Page 12: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

FRONT

BACKTOP

Page 13: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

FRONT

BACKTOP

BOTTOM

Page 14: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

FRONT

BACKTOP

BOTTOM

LEFT

Page 15: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This rectangular prism has 6 faces.

FRONT

BACKTOP

BOTTOM

LEFT

RIGHT

Page 16: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

BACK

This rectangular prism has 6 faces.

FRONT

TOP

BOTTOM

LEFT

RIGHT

Page 17: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

BACK

This rectangular prism has 6 faces.

FRONT

BOTTOM

LEFT

RIGHT

TOP

Page 18: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

BACK

This rectangular prism has 6 faces.

FRONT

BOTTOM

LEFT

RIGHT

TOP

Page 19: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

BACK

This rectangular prism has 6 faces.

FRONT

BOTTOM

LEFT

RIGHT

TOP

Page 20: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

BACK

This rectangular prism has 6 faces.

FRONT

BOTTOM

LEFT

RIGHT

TOP

Page 21: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

This square pyramid has 5 faces.

The faces consist of 4 triangles and a square.

Page 22: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

The faces consist of 4 triangles and a square.

Page 23: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 faces.

Page 24: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 faces.

Page 25: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 faces.

Page 26: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 faces.

Page 27: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 faces.

Page 28: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 faces.

Page 29: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

An edge is a line segment where two faces meet.

Page 30: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A rectangular prism has 12 edges.

Page 31: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 6 edges.

Page 32: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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

more edges.

Page 33: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A vertex is a fancy word for “corner.”

Every triangle has 3 vertices (corners).Points A, B, and C are vertices.

A B

C

Page 34: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A rectangular prism has 8 vertices.

Page 35: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A rectangular prism has 8 vertices.

Page 36: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 vertices.

Page 37: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

A triangular pyramid has 4 vertices.

Page 38: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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

Page 39: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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 5 8

Tri. Prism 5 6 9

Page 40: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Euler looked for a relationship between these numbers.

Polyhedron # of Faces # of Vertices # of Edges

Cube 6 8 12

Sq. Pyramid 5 5 8

Tri. Prism 5 6 9

Page 41: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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 5 8

Tri. Prism 5 6 9

Page 42: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

Faces + Vertices –2 = Edges

Polyhedron # of Faces # of Vertices # of Edges

Cube 6 8 12

Sq. Pyramid 5 5 8

Tri. Prism 5 6 9

+ - 2 =

+ - 2 =

+ - 2 =

Page 43: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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 5 8

Tri. Prism 5 6 9

Pent. Prism 7 10

+ - 2 =

+ - 2 =

+ - 2 =

Page 44: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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 5 8

Tri. Prism 5 6 9

Pent. Prism 7 10

+ - 2 =

+ - 2 =

+ - 2 =

+ - 2 =

Page 45: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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 5 8

Tri. Prism 5 6 9

Pent. Prism 7 10 15

+ - 2 =

+ - 2 =

+ - 2 =

+ - 2 =

Page 46: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

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.

Page 47: Euler’s Formula A Naturally Occurring Function. Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”

THE END!


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