Regular Polyhedra and the Platonic Solids The best known of all the polyhedra are the regular polyhedra, or Platonic solids. A regular polyhedron is a convex polyhedron whose faces are congruent regular polygons, the same number of which meet at each vertex. The ancient Greeks proved that there are only five regular polyhedra. o The tetrahedron has equilateral triangles for faces o The cube has square faces o The octahedron has equilateral triangular faces o The dodecahedron has pentagons for faces o The icosahedron has equilateral triangular faces Platonic Solids in Nature Semiregular Polyhedra Some polyhedra have two or more different types of regular polygons for faces. Polyhedra whose faces are two or more regular polygons with the same arrangement of polygons around each vertex are called semiregular polyhedra. Task 0—Practice while you work Refer to your “Practicing with Practices” pages and write notes whenever you use a practice while working today. We’ll share after we do our math tasks. Task 1-Platonic Solids: Use the Polydrons and build a set of Platonic Solids Paper nets for the Platonic Solids are linked to your course Moodle page Task 2—Relationship between Vertices, Edges and Faces: In the table, record the numbers of vertices, faces, and edges for each of the Platonic solids you constructed. Vertices (V) Edges (E) Faces (F) Tetrahedron Cube Octahedron Dodecahedron Icosahedron There is a relationship among the numbers of vertices, faces, and edges. Make and record a conjecture about this relationship and express it in terms of the letters V, F, and E. Conjecture: Task 3—Truncation Station: If you snip (truncate) the corners (vertices) of the Platonic Solids, you make a new set of semiregular polyhedra. a. Use the Polydrons and build each of the following semiregular polyhedra (if you can, or note that you can’t). Note that the scale changes when you use the Polydrons. b. Identify the truncated shape (truncated cube, truncated tetrahedron, etc.). Note that you can snip the vertex “a little way down” or halfway done and get different results. c. Test your conjecture from Task 2. Does it work? Task 4: Surface Area of a Sphere—We will try to determine the formula for the surface area of a sphere using a symmetric piece of citrus fruit. a. Determine the radius of your piece of fruit. This will be an estimate, since your fruit is not perfectly spherical. b. Set the compass opening to the radius of your fruit and draw five circles with the same radius as your fruit on a blank piece of paper. c. Peel your piece of fruit, breaking the peel into small pieces. See how many of the circles can be filled with the pieces. Be sure to leave as little space between the pieces as possible and to use very flat pieces (break them smaller if they are not flat). d. You can now see how many circular areas can be covered by the surface (skin) of our piece of spherical fruit. The surface area of a sphere of radius r is _________________________. Task 5: Volume—Cone, Cylinder, Square Prism, Cube and Sphere a. Measure the polyhedra using the centimeter measuring tapes. Measure the "parts" of the polyhedral models, as accurately as you can (in cm), that you will need to compute the volume of each polyhedron. Polyhedron Needed measurements (in centimeters) Cone Cylinder Square Pyramid Cube Sphere b. Compare the polyhedra using your measurements and visual inspection. What relationships for volume do you think there are? c. Measure the polyhedral volumes with water, what relationships for volume can you find?