Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
MATH 113
Section 8.3: Three-Dimensional Figures
Prof. Jonathan Duncan
Walla Walla University
Winter Quarter, 2008
Conclusion
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Outline
1
Polyhedra
2
Prisms and Pyramids
3
Other Three Dimensional Figures
4
Conclusion
Conclusion
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Definitions
The three-dimensional version of a polygon is called a polyhedron.
Polyhedra
A polyhedron is a simple closed surface composed entirely of
polygon regions.
Example
Are all of the following polyhedra?
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Parts of a Polyhedra
Just as with polygons, we can name the various parts of polyhedra.
Parts of a Polyhedron
The surface of a polyhedron can be separated into various parts.
Face
A face is one of the polygon regions on the surface of the
polyhedron.
Edge
An edge is a line segment connecting two adjacent polygon regions.
Vertex
A vertex is a point connecting three or more edges.
Example
Sketch a cube and identify its faces, edges, and vertices. What can
you say about the faces of the cube?
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Connecting Polyhedra with Polygons
We know several terms from our study of polygons. Can these be
applied to polyhedra as well?
Example
Decide if each term or concept below can be generalized to
polyhedra and if it can, describe how.
Simple Closed Curve
Sides
Interior Angles
Classification by the Number of Sides
Convex and Concave
Regular
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Regular Polyhedra
Just as we defined regular polygons in two dimensions, we can
define regular polyhedra in three.
Regular Polyhedra
A regular polyhedron is a convex polyhedron in which the faces are
congruent regular polygons and in which the number of edges
which meet at each vertex are the same.
Example
What is the simplest regular polyhedron you can think of?
Example
How many regular polygons are there? How many regular
polyhedra are there?
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
The Eight Platonic Solids
You may find it surprising that while there are infinitely many
regular polygons, there are only 8 regular polyhedra.
The Platonic Solids
The eight regular polyhedra are often called the Platonic solids,
named for the ancient Greek philosopher Plato who theorized that
the classical elements were constructed from these regular solids.
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Kepler’s Model of the Solar System
These eight shapes held such meaning for early mathematicians and
scientists that Johannes Kepler attempted to find a relationship between
the five known planets during his time and the five Platonic solids.
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Euler’s Discovery
Below is a table with the number of vertices, edges, and faces in
the regular polyhedra.
Vertices, Edges, and Faces
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Vertices
4
8
6
20
12
Edges
6
12
12
30
30
Faces
4
6
8
12
20
Euler’s Formula
The relationship between these values is given by Euler’s formula:
V + F = E + 2. This works for all polyhedra.
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
What is a Prism
Just as we could divide polygons into families based on the number
of sides, we can divide polyhedra into families based on other
characteristics.
Prisms
A polyhedron with two parallel bases what are congruent polygons
is called a prism. The non-base faces are called lateral faces.
Example
What one shape can be used the describe the lateral faces of all
prisms?
Prisms as Stacks of Polygons
Another way to think of a prism is as many copies of the same
polygon stacked on top of one another.
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Types of Prisms
There are many different types of prisms. They can be classified in
several ways.
Clarification by Base
Prisms can be classified by the type of polygon used as a base. For
example, a triangular prism or a rectangular prism.
Classification by Lateral Faces
Prisms can be classified by their type of lateral faces. Prisms with
rectangular lateral faces are called right prisms. Prisms which are
not right prisms are called oblique prisms.
Example
Sketch a right pentagonal prism and an oblique rectangular prism.
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
Pyramids
Another interesting type of polyhedron is a pyramid. This term is
broader than what is usually thought of as a pyramid.
Pyramids
A pyramid is a polyhedron whose base is a polygon and whose
other faces are triangles with a common vertex. That common
vertex is called the apex of the pyramid.
Pyramids and Stacking
Alternatively, a pyramid can be built by stacking smaller and
smaller similar polygons.
Example
How do pyramids and prisms compare. In particular think about: bases
and right vs. oblique pyramids and prisms.
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
From Prisms to Cylinders
If we think of taking right prisms with n-gon bases and letting n
increase, the shape we approach is called a cylinder.
Cylinders
A cylinder is a simple closed surface that is bounded by two
congruent circles that lie in parallel planes.
Example
What aspects or terms used to describe prisms can also be used to
describe cylinders?
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Conclusion
From Pyramids to Cones
If we think of making pyramids with n-gon bases and letting n
increase, we approach a shape called a cone.
Cones
A cone is the collection of all line segments between a simple
closed region of a plane (the base) and a point outside the plane
(the apex).
Example
By taking cross-sections of a right-circular cone, we can get several
different two-dimensional shapes called conic sections. These are:
Circle
Ellipse
Parabola
Hyperbola (single side)
Polyhedra
Prisms and Pyramids
Other Three Dimensional Figures
Important Concepts
Things to Remember from Section 8.3
1
Definition and parts of polyhedra
2
Euler’s Formula for Polyhedra
3
Definition and types of prisms
4
Definition and types of pyramids
5
Definitions and types of cones and cylinders
Conclusion