NAME
1-7
DATE
PERIOD
Enrichment
Polyhedrons
1
As you know, any three noncollinear points determine a unique
plane. Consider what happens with four noncoplanar points.
1. How many unique planes can be determined by four
noncoplanar points?
2
4
2. Drawing all of these planes forms a closed polyhedron.
Classify this polyhedron by name.
3
Now consider a set of n points such that no more than three points in this set
are coplanar.
3. If every set of three points determines a unique plane and these planes are
drawn, the result will be a closed polyhedron. What will be the shape of each of
the faces of such a polyhedron?
4. Complete the table to show the number of
faces of the resulting polyhedron for each
number of noncoplanar points (column 2).
Points
Faces
(column 2)
Edges
(column 3)
5
5. Extend your pattern to find each of
the following.
6
7
a. the number of triangular faces formed
using a set of 20 noncoplanar points
8
b. the number of points necessary to form a polyhedron of 20 faces
6. Generalize your pattern to write the expression representing the number
of triangular faces formed using a set of n noncoplanar points.
7. Determine whether your formula is true for each of the polyhedrons
formed in this activity. If so, fill in the number of edges for each polyhedron
in the table (column 3).
Chapter 1
48
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4