University of Arizona
Institute for Mathematics and Education

September 21, 2010

Platonic Solids- How they really relate presented by Ron Hopley


Ron introduced his presentation by showing a nested set of the Platonic Solids. He showed his Acrospin files to describe the mathematical relationships that occur in various nested configurations. Much of the information can be found in his article, Nested Polyhedra Project published in Mathematics Teacher , May 1994.



Participants were given an icosahedral die (found in game stores). They measured an edge length of the die. Using that length, they calculated the length needed for the octahedron. Using rulers and compasses, they constructed the net for the octahedron and glued the necessary tabs. Once formed, the icosahedral die nested in the octahedron.


Using the edge length of the octahedron, the edge length for the tetrahedron was calculated. The tetrahedron net was constructed. Once formed, the octahedron nested nicely in the tetrahedron.


After dinner, the tetrahedron edge length was used to calculate the cube edge length since a tetrahedron edge will become a diagonal of a cube face. Once created the tetrahedron nested in the cube.


The cube edge represents a diagonal of a pentagonal face of the dodecahedron. Using trigonometry, we were able to calculate the edge length for the dodecahedron. The creation of the dodecahedron was left for "homework".


This evening's mathematical adventures addressed numerous performance and process standards were utilized in the presentation. Some of these include:

  • MHS-S3C3-01 Create and explain the need for equivalent forms of an equation or expression.
  • MHS-S4C1-02 Visualize solids and surfaces in 3-dimensional space when given 2-dimensional representations and create 2-dimensional representations for the surfaces of 3-dimensional objects.
  • MHS-S4C1-06 Solve problems using angle and side length relationships and attributes of polygons.
  • MHS-S4C1-10 Solve problems using right triangles, including special triangles.
  • MHS-S4C1-11 Solve problems using the sine, cosine, and tangent ratios of the acute angles of a right triangle.
  • MHS-S4C4-05 Calculate the surface area and volume of 3-dimensional figures and solve for missing measures.



    Ron Group Mike
    Ron Hopley discusses the relationship between a cube edge and the pentagon's diagonal.A group works on the net construction.Mike Schmidt shows Daniel Schneider how well his tetrahedron fits in his cube.