Constructing Polyhedra and Developing Euler’s Formula

Cooperative Learning Lesson Plan designed by Rebecca A. Zeier



To move beyond the two-dimensional world of geometry into the three-dimensional world of geometry.  In this lesson we will

explore faces, edges, and vertices of Pythagoras’ five regular solids called Polyhedra.  From these five polyhedra we will use

the numbers of faces, edges, and vertices to discover Euler’s formula of F + V = E + 2.  Appropriate for any level of geometry.



Patterns and directions for all five regular polyhedra will be provided.  In addition to the patterns, students will be given

scissors and tape to put together the polyhedra.



(assuming 20 students in class)  Groups will be made up of 5 groups of 4 students.  I will assign groups randomly using playing



Group Interdependence:

Students will be assigned roles.  The following roles include the “cutter”, the “taper”, the “put it together-er”, and the “counter”.  I will also hand out a chart for all members to fill out with their polyhedron’s number of faces, edges, and vertices (see attached).  One student will be randomly chosen to show their polyhedron and give results of their counting.


Individual Accountability:

Students will be assigned roles (see group interdependence for role assignments).  Each student will be accountable for helping their group members assemble on of the five regular polyhedra.  I will give each group the pattern necessary (see attached) to construct their polyhedron.  Individuals in each group will be responsible for cutting out the pattern, putting the pattern together, taping together the pattern, and counting the faces, edges and vertices.  Each individual student will be responsible for recording these numbers in the chart given to them (see attached).  Instructed to compute F + V – E, they should get the number 2.  Using the RAND function on the calculator, I will randomly select one student from each group to share their polyhedron and their results.  As a class (see closure below) we will discover that the result of this computation is always 2 and will develop Euler’s formula of F + V = E + 2.


Check for Comprehension:

Comprehension of Euler’s formula will be revealed through homework (see attached).


End Product:  

All groups will have constructed one of the five regular polyhedra. Instructed to compute

F + V – E, students will discover that Euler’s number is 2.



Each group will present their polyhedron stating the number of faces, edges, and vertices their polyhedron has.  The class should see that the computation of F + V – E will always equal 2.   From this, they will be able to develop Euler’s formula for polyhedra.