Lesson 9 — The Regular Polyhedra
Big idea: There are exactly five regular polyhedra, called the Platonic solids, each made of congruent regular polygons with the same number of faces meeting at every vertex.
Words to know
| Word | Meaning |
|---|---|
| Regular polyhedron | A convex solid whose faces are congruent regular polygons and that has an equal number of polygons meeting at each vertex. |
| Tetrahedron | A regular polyhedron with 4 equilateral-triangle faces. |
| Octahedron | A regular polyhedron with 8 equilateral-triangle faces. |
| Icosahedron | A regular polyhedron with 20 equilateral-triangle faces. |
| Dodecahedron | A regular polyhedron with 12 regular-pentagon faces. |
| Cube (regular hexahedron) | A regular polyhedron with 6 square faces. |
Concepts to know
- A regular polyhedron must be convex, have congruent regular polygon faces, and have the same number of faces at each vertex.
- There are exactly five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
- Polyhedra are named by their number of faces: tetra- = 4, octa- = 8, dodeca- = 12, icosa- = 20.
- Three regular polyhedra have equilateral-triangle faces: tetrahedron (4), octahedron (8), icosahedron (20).
- The cube has 6 square faces; the dodecahedron has 12 regular-pentagon faces.
- The regular polyhedra are also called the Platonic solids.
- Faces meeting at each vertex: tetrahedron 3, cube 3, octahedron 4, dodecahedron 3, icosahedron 5.
- The Euler characteristic F + V − E equals 2 for each of these solids (review exercise).
Formulas / rules
- No new area or volume formula in this lesson.
- Euler characteristic: F + V − E = 2
- Face counts: 4, 6, 8, 12, 20.
- To check a candidate: faces must be congruent regular polygons and the same number must meet at every vertex.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Name a polyhedron from its faces | Count the faces and use the prefix: 4 tetra, 6 hexa, 8 octa, 12 dodeca, 20 icosa. |
| Tell whether a solid is regular | Check convexity, congruent regular polygon faces, and the same number of faces at each vertex. |
| Count vertices or edges | Use a model or the review table; then check F + V − E = 2. |
| Identify the faces of a regular polyhedron | Triangles for tetrahedron, octahedron, icosahedron; squares for the cube; pentagons for the dodecahedron. |
| Find how many faces meet at a vertex | Tetrahedron 3, cube 3, octahedron 4, dodecahedron 3, icosahedron 5. |
| Relate a cube to an octahedron | The octahedron’s vertices are at the centers of the cube’s faces. |
Watch out
- “Regular” requires more than congruent faces; the same number of faces must meet at every vertex.
- There are only five regular polyhedra — no regular solid with hexagonal faces exists.
- The dodecahedron has pentagon faces, not hexagons.
- A cube is a regular hexahedron, but not every hexahedron is regular.
- Do not confuse the number of faces with the number of sides of each face.