Five Regular Geosolids Meeting
(3-D Geometry)
provided by Pitsco Education, Inc.
Topic
Using the straws and pipe cleaners provided in the Club in a Box Resource Kit, students will
explore five regular polyhedra and Euler’s Theorem (v + f – e = 2).
Materials Needed
♦ PITSCO supplies (straws and pipe cleaners) provided in the Club in a Box Resource Kit
♦ Copies of the Student Activity Sheet (shown on page 27 and available in the PITSCO
supplies kit or online at www.mathcounts.org on the MCP Members Only page of the Club
Program section)
Meeting Plan
OVERVIEW
Solid geometry is best done in three dimensions as it allows students to experience geometric
solids as they really are – three dimensional. In this activity, students use straws and pipe
cleaners (chenille stems) to construct wire frames of each of the five regular polyhedra:
tetrahedron, cube, octahedron, dodecahedron and icosahedron.
MATH CONNECTIONS
Students learn and experience the terminology and features of regular geometric solids by
constructing these polyhedra. The terms vertex, edge and face become clear as they are
discussed in the context of constructing these solid figures. (Before starting to construct the
figures, it may be helpful to review with your students the vocabulary listed on the next page.)
Euler’s Theorem (v + f – e = 2) can be explored by students as they (1) determine the number
of vertices (v), faces (f) and edges (e); (2) record those values in a table; and (3) verify the
calculation for each of the polyhedra. The theorem also can be used to determine the value of
an unknown variable if the other two variables are given.
CONSTRUCTION DETAILS
Students will refer to the Basic Construction Techniques section
in the User Guide (included in the PITSCO supplies). This will
provide information about constructing three-, four- and five-legged
connectors (vertices). Video versions of these techniques also can
be viewed at www.zoonzone.com/video.asp?actid=8.
Basic instructions explaining how to build each geometric solid also
are provided in the User Guide. By looking at the instructions and
figures provided, students can see how the solids should appear
when they are finished and can determine by trial and error how to
complete the construction.
Forming a three-legged connector
FACILITATING THE ACTIVITY
The following is based on a group of 10 students. You can modify
it to fit your club size and needs. Be sure to consider the number of
straws (100) and pipe cleaners (100) available. Each group (or each
student) will have a copy of the Student Activity Sheet to complete.
2009–2010 MATHCOUNTS Club Resource Guide
Forming a five-legged connector
25
1. Have five students (Group A) each work on constructing a tetrahedron
(Activity 1 with questions), and have five students (Group B) each work on
constructing a cube (Activity 2 with questions).
2. When they have completed the construction and answered the questions for
their respective activities, have the two groups trade structures and complete
the questions for those structures.
3. Have the students disassemble all but one of each structure (so there is one tetrahedron and
one cube left intact). The straws and pipe cleaners from the disassembled structures, along with
the unused straws and pipe cleaners can be used in the following constructions.
4. Group A should construct an octahedron and a dodecahedron (Activities 3
& 4) and answer the questions for the activities. Group B should construct an
icosahedron (Activity 5) and answer the questions for the activity.
5. When the questions for each structure are answered, the two groups should
trade structures and answer the questions for those structures.
VOCABULARY
angle – a figure composed of two lines or rays with a common end point (the vertex of the angle)
area – the size of an enclosed region
edge – the set of points where the faces of a geometric solid intersect
face – a planar surface of a geometric solid
polyhedra – the plural of polyhedron, a three-dimensional figure with many faces
solid geometry – the branch of mathematics dealing with three-dimensional shapes
surface area – for a polyhedron, the sum of the area of all the faces
tetrahedron – a polyhedron with four triangular faces
three-dimensional (3-D) – having length, width and depth
vertices – the plural of vertex; the set of all the points of intersection of a geometric solid’s edges
volume – the amount of space occupied by a three-dimensional object
Answers to Student Activity Sheet: Activity 1: 4, 6, 4, equilateral triangle; Activity 2: 8, 12, 6, square;
Activity 3: 6, 12, 8, equilateral triangle; Activity 4: 20, 30, 12, regular pentagon; Activity 5: 12, 30, 20,
equilateral triangle
Possible Next Steps
Once students have constructed all five of the structures, they can be challenged to do the
following:
• Calculate the angle measure between the parts of each of the five geometric solids.
• Calculate the surface area of each of the five geometric solids.
• Calculate the volume of the five geometric solids.
• Research other three-dimensional geometric shapes on the Internet or in books, and construct
them using straws and pipe cleaners. Determine the number of edges, the shape of the faces,
and the number of faces for each shape. Calculate the angle measure between the edges, the
surface area, and the volume of the figures.
26 2009–2010 MATHCOUNTS Club Resource Guide
Five Regular Geometric Solids
Student Activity Sheet
Structures that have length, width, and height are
called three-dimensional figures or structures.
These figures have edges, faces, and vertices.
Edges are the sides of the structure (the straws in
our structures). Faces are the geometric shapes
that the edges form. Vertices are the points at
which the edges of the structure intersect.
In a regular geometric solid, all the edges have
equal lengths and all the faces are the same
shape. There are only five structures possible
that are regular geometric solids: tetrahedron,
hexahedron (cube), octahedron, dodecahedron,
and icosahedron.
In the following activities, refer to the Basic
Construction Techniques to see how to make the
various vertices.
Activity 1
Make four of the three-legged vertices, and use straws to connect them to form a
pyramid shape. The mathematical name for this structure is a tetrahedron.
a. Count the number of vertices. _____
b. Count the number of edges the structure has. _____
c. Count the number of faces. _____
d. What is the name of the shape of these faces? ______________
Activity 2
Construct a cube using three-legged vertices and straws to construct. Another
name for a cube is a hexahedron.
a. Count the number of vertices. _____
b. Count the number of edges the structure has. _____
c. Count the number of faces. _____
d. What is the name of the shape of these faces? _______________
Activity 3
Construct a octahedron using four-legged vertices and straws.
a. Count the number of vertices. _____
b. Count the number of edges the structure has. _____
c. Count the number of faces. _____
d. What is the name of the shape of these faces? _______________
Activity 4
Construct a dodecahedron using three-legged vertices and straws.
a. Count the number of vertices. _____
b. Count the number of edges the structure has. _____
c. Count the number of faces. _____
d. What is the name of the shape of these faces? _______________
Activity 5
Construct a icosahedron using five-legged vertices and straws.
a. Count the number of vertices. _____
b. Count the number of edges the structure has. _____
c. Count the number of faces. _____
d. What is the name of the shape of these faces? _______________
2009–2010 MATHCOUNTS Club Resource Guide
27