Geometry
Polyhedra
Goals
Know terminology about solids.
 Identify solids by type.
 Use Euler’s Theorem to solve problems.

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Polyhedron



A solid that is bounded
by polygons.
The polygons are
faces.
An edge is the
intersection of two
faces.
A vertex is the
intersection of three or
more faces.
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Face
Face
Face

3
Polyhedron
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Polyhedron Views
Solid
Wire Frame
All three views will be used
in these presentations, the
text and other materials.
Hidden Line
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Which of these are Polyhedrons?
NO
YES
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YES
NO
YES
6
Concave Polyhedra
A diagonal, or part of a
diagonal, is outside the figure.
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Regular Polyhedra

All of the faces are
congruent, regular
polygons.
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Cross Section

The intersection of a solid and a plane.
Cross section is a circle.
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Cross Section

What is the
intersection now?
Cross section is a rectangle.
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What would the cross section be?
A Square
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Leonard Euler



1707 – 1783
Probably the greatest
mathematician of all
time.
Worked in, and made
enormous
contributions to, every
branch of
mathematics.
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Euler’s Formula
Count F, the
number of faces.
12
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4
3 5
6 F=6
13
Euler’s Formula
7
8
Count V, the
number of
vertices.
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2
1
5
6
3
4
V=8
14
Euler’s Formula7
6
9
Count E, the
number of edges.
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10 5
3 12
2
1
8 11
4
E = 12
15
Euler’s Formula
Faces =
6
Vertices =
8
V+F=E+2
Edges =
12
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Euler’s Formula
Faces =
6
Vertices =
8
6 + 8 = 12 + 2
Edges =
12
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Euler’s Formula
Faces =
6
Vertices =
8
6 + 8 = 12 + 2
14 = 14
Edges =
12
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Euler’s Formula
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Try another figure…
Faces =
Vertices =
Edges =
F+V=E+2
5+5=8+2
10 = 10
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Euler’s Formula
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Solve:
A polyhedron has 8 faces and 12 vertices.
How many edges does it have?
 18
V+F=E+2
 12 + 8 = E + 2
 20 = E + 2
 E = 18

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Solve:
A polyhedron has 24 vertices and 36
edges. How many faces does it have?
 14
V+F=E+2
 24 + F = 36 + 2
 24 + F = 38
 F = 14

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Solve:
A polyhedron has 32 faces and 60 edges.
How many vertices does it have?
 30
V+F=E+2
 V + 32 = 60 + 2
 V + 32 = 62
 V= 30

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The Platonic Solids
There are only five of them.
 They are regular, convex polyhedra.
 First described ca. 350 BC by Plato in
Timaeus.
 Have been found in many ancient cultures.

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The Five Platonic Solids
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Tetrahedron
Has four triangular sides.
Associated with fire.
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Hexahedron (cube)
Has six square sides.
Associated with earth.
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Octahedron
Has eight triangular sides.
Associated with air.
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Dodecahedron
Has 12 pentagonal faces.
Associated with the heavens.
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Icosahedron
Has 20 triangular faces.
Associated with water.
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Johannes Kepler

In 1596 Kepler published
a tract called The Cosmic
Mystery in which he
envisioned the universe
as consisting of nested
Platonic Solids whose
inscribed spheres
determine the orbits of
the planets, all enclosed
in a sphere representing
the outer heavens.
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Dungeons and Dragons
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Public Toilets in South Korea
This is not a
Platonic Solid. It is a
compound
polyhedron. Can you
find out its correct
name?
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Platonic Solid Links
Mathworld
GSP Icosahedron
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Summary
A polyhedron is a solid object.
 The sides are faces.
 Regular polyhedra have congruent faces.
 There are 5 regular polyhedra (the
Platonic Solids).
 Euler’s Formula: F + V = E + 2

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Homework
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