2.11 Surface Area and Volume
The Polyhedron
Polygon or Polyhedron –A polyhedron is defined using polygons, so in the beginning students will understand the
difference. After some time has passes though, students tend to get these similar sounding words confused. Remind
them that polygons are two-dimensional and polyhedrons are three-dimensional. The extra letters in polyhedron
represents it spreading out into three-dimensions.
The Limitations of Two Dimensions –It is difficult for students to see the two-dimensional representations of
three-dimensional figures provided in books and on computer screens. A set of geometric solids is easily obtained
through teacher supply companies, and are extremely helpful for students as they familiarize themselves with three-
dimensional figures. When first counting faces, edges, and vertices most students need to hold the solid in their
hands, turn it around, and see how it is put together. After they have some experience with these objects, student
will be better able to read the figures drawn in the text to represent three-dimensional objects.
Assemble Solids –A valuable exercise for students as they learn about polyhedrons is to make their own. Students
can cut out polygons from light cardboard and assemble them into polyhedrons. Patterns are readily available. This
hands-on experience with how three-dimensional shapes are put together will help them develop the visualization
skills required to count faces, edges, and vertices of polyhedrons described to them.
Computer Representations –When shopping on-line it is possible to “grab” and turn merchandise so that they can
be seen from different perspectives. The same can be done with polyhedrons. With a little poking around students
can find sights that will let them virtually manipulate a three-dimensional shape. This is another possible option to
develop the students’ ability to visualize the solids they will be working with for the remainder of this chapter.
Using the Contrapositive –If students have already learned about conditional statements, point out to them that
Example Six in this section makes use of the contrapositive. Euler’s formula states that if a solid is a polyhedron,
thenv+f=e+2. The contrapositive is that ifv+f 6 =e+2, then the solid is not a polyhedron. Students need
periodic review of important concepts in order to transfer them to their long-term memory. For more review of
conditional statements, see the second section of the second chapter of this text.
Representing Solids
Each Representation Has Its Use –Each of the methods for making two-dimensional representations of three-
dimensional figures was developed for a specific reason and different representation are most appropriate depending
on what aspect of the geometric solid is of interest.
Perspective – used in art, and when one wants to make the representation look realistic
Isometric View – used when finding volume
Orthographic View – used when finding surface area
Cross Section – used when finding volume and the study of conic sections (circles, ellipses, parabolas, and hyperbo-
las) is based on the cross sections of a cone
Nets – used when finding surface area or assembling solids
2.11. Surface Area and Volume