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**A Naturally Occurring Function**

Euler’s Formula A Naturally Occurring Function

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**Leonhard Euler was a brilliant Swiss mathematician**

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

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

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Poly-what?

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

A polyhedron is a 3 dimensional shape with flat sides.

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

PRISMS PYRAMIDS

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**Any polyhedron has faces, vertices, and edges.**

VERTEX

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A face is a flat side.

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**This rectangular prism has 6 faces.**

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**This rectangular prism has 6 faces.**

FRONT

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**This rectangular prism has 6 faces.**

BACK FRONT

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**This rectangular prism has 6 faces.**

TOP BACK FRONT

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**This rectangular prism has 6 faces.**

TOP BACK FRONT BOTTOM

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**This rectangular prism has 6 faces.**

TOP BACK FRONT LEFT BOTTOM

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**This rectangular prism has 6 faces.**

TOP BACK FRONT RIGHT LEFT BOTTOM

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**This rectangular prism has 6 faces.**

TOP BACK FRONT RIGHT LEFT BOTTOM

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**This rectangular prism has 6 faces.**

TOP BACK FRONT RIGHT LEFT BOTTOM

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**This rectangular prism has 6 faces.**

TOP LEFT BACK FRONT RIGHT BOTTOM

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**This rectangular prism has 6 faces.**

TOP LEFT BACK FRONT RIGHT BOTTOM

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**This rectangular prism has 6 faces.**

TOP RIGHT LEFT BACK FRONT BOTTOM

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**This square pyramid has 5 faces.**

The faces consist of 4 triangles and a square.

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**The faces consist of 4 triangles and a square.**

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**A triangular pyramid has 4 faces.**

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**A triangular pyramid has 4 faces.**

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**A triangular pyramid has 4 faces.**

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**A triangular pyramid has 4 faces.**

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**A triangular pyramid has 4 faces.**

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**A triangular pyramid has 4 faces.**

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**An edge is a line segment where two faces meet.**

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**A rectangular prism has 12 edges.**

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**A triangular pyramid has 6 edges.**

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

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**A vertex is a fancy word for “corner.”**

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

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**A rectangular prism has 8 vertices.**

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**A rectangular prism has 8 vertices.**

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**A triangular pyramid has 4 vertices.**

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**A triangular pyramid has 4 vertices.**

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

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**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

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**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

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**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

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**Faces + Vertices –2 = Edges**

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

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**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 = +

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**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 = +

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**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 = +

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**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.

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THE END!

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