euler’s formula a naturally occurring function. leonhard euler was a brilliant swiss...
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Euler’s Formula
A Naturally Occurring Function
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Leonhard Euler was a brilliant Swiss mathematician. He is
often referred to as the “Beethoven of Mathematics.”
![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.”](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d015503460f949d3d6b/html5/thumbnails/3.jpg)
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.
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All prisms and pyramids are examples of polyhedra (plural for
polyhedron).
POLYHEDRA
PRISMS PYRAMIDS
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Any polyhedron has faces, vertices, and edges.
EDGE
FACE
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.
FRONT
BACK
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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.”](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d015503460f949d3d6b/html5/thumbnails/13.jpg)
This rectangular prism has 6 faces.
FRONT
BACKTOP
BOTTOM
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This rectangular prism has 6 faces.
FRONT
BACKTOP
BOTTOM
LEFT
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This rectangular prism has 6 faces.
FRONT
BACKTOP
BOTTOM
LEFT
RIGHT
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BACK
This rectangular prism has 6 faces.
FRONT
TOP
BOTTOM
LEFT
RIGHT
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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.”](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d015503460f949d3d6b/html5/thumbnails/18.jpg)
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.”](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d015503460f949d3d6b/html5/thumbnails/19.jpg)
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.”](https://reader035.vdocument.in/reader035/viewer/2022062714/56649d015503460f949d3d6b/html5/thumbnails/20.jpg)
BACK
This rectangular prism has 6 faces.
FRONT
BOTTOM
LEFT
RIGHT
TOP
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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.”
Every triangle has 3 vertices (corners).Points A, B, and C are vertices.
A B
C
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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 5 8
Tri. Prism 5 6 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 5 8
Tri. Prism 5 6 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 5 8
Tri. Prism 5 6 9
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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 =
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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 5 8
Tri. Prism 5 6 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 5 8
Tri. Prism 5 6 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 5 8
Tri. Prism 5 6 9
Pent. Prism 7 10 15
+ - 2 =
+ - 2 =
+ - 2 =
+ - 2 =
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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!