2.1.7 Platonic Solids and Euler’s Formula
The five Platonic solids are regular, convex polyhedra with congruent faces. Understanding their properties is essential for number sense problems.
The Five Platonic Solids
| Name | Faces | Vertices | Edges | Face Shape |
|---|---|---|---|---|
| Tetrahedron | 4 | 4 | 6 | Triangle |
| Hexahedron (Cube) | 6 | 8 | 12 | Square |
| Octahedron | 8 | 6 | 12 | Triangle |
| Dodecahedron | 12 | 20 | 30 | Pentagon |
| Icosahedron | 20 | 12 | 30 | Triangle |
Euler’s Formula
For any convex polyhedron:
Where:
- = number of vertices
- = number of edges
- = number of faces
Problem Set 2.1.7
A dodecahedron has ___ vertices
An icosahedron has ___ congruent faces
The area of the base of a tetrahedron is 4 ft². The total surface area is ___ ft²
A tetrahedron has ___ vertices
An octahedron has ___ edges
A hexahedron has ___ faces
A dodecahedron is a platonic solid with 30 edges and ___ vertices
An octahedron has ___ vertices
An icosahedron is a platonic solid with 30 edges and ___ vertices
A dodecahedron is a platonic solid with 30 edges and ___ vertices