Geometry
11.4 Three-Dimensional Figures
mbhaub@mpsaz.org
Warmup
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Geometry 11.4 Three-Dimensional
Figures
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Essential Question

What is the relationship between the
numbers of vertices V, edges E, and faces F
of a polyhedron
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Geometry 11.4 Three-Dimensional
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Goals
Classify solids
 Describe cross sections
 Sketch and describe solids of revolution

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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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Geometry 11.4 Three-Dimensional
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Face
Face
Face

5
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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Types of Solids
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Naming Polyhedra
To name a prism or a pyramid, use the shape of
the base. The two bases of a prism are congruent
polygons in parallel planes. The base of a
pyramid is a polygon.
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Name each solid.
Which of these are Polyhedrons?
Triangular
Prism
Cylinder
NO
YES
YES
Sphere
NO
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Pentagonal
Prism
Square
Pyramid
YES
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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.
The cross
section is
parallel to
the bases.
Cross section is a circle.
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Cross Section

What is the
intersection now?
Cross section is a rectangle.
The cross section is
perpendicular to the
bases.
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What would the cross section be?
The cross
section is
parallel to
the base.
A Square
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What would the cross section be?
The cross section is
perpendicular to both
bases.
A Pentagon
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What would the cross section be?
The cross section is
slanted and intersects
both bases.
A Trapezoid
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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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A solid of revolution is a three-dimensional figure
that is formed by rotating a two-dimensional shape
around an axis.
If you rotate a rectangle around one of its sides, the
path it makes through space is a cylinder.
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Solids of Revolution
Example: Sketching a Solid of Revolution
Draw the solid of revolution formed by the shape
rotated around the axis given. Describe the
resulting shape.
A cylinder with the center removed.
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Check it Out! Example 1
Draw the solid of revolution formed by the shape
rotated around the axis given. Describe the
resulting shape.
A donut shape, also known as a torus.
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Example 2: Sketching a Solid of Revolution
Draw a two-dimensional shape and axis of
rotation that could form each figure.
Find the axis of
symmetry.
The two-dimensional shape
should match the outline of
one side of the shape.
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Check it Out! Example 2
Draw a two-dimensional shape and axis of
rotation that could form the sports drink bottle.
Find the axis of
symmetry.
The two-dimensional shape
should match the outline of
one side of the shape.
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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:

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