Polyhedrons are FUN. If that weren't true, nobody would play with hexahedrons (otherwise known as dice). Or Minecraft. On high school tests last year, this concept showed up 10 times, with a median placement at question # 47.
Number Dojo Level: 195
Think of a polyhedron as a 3-dimensional polygon. A regular polyhedron is one in which each side (or face) is a regular polygon, identical in size and shape. For number sense, we only need to be concerned about the 5 different regular polyhedrons (known as the Platonic solids):
6 faces: Hexahedron (cube)
8 faces: Octahedron
12 faces: Dodecahedron
20 faces: Icosahedron
You'll need to know Euler's Formula, which is:
- V represents the # of vertices,
- E represents the number of edges, and
- F represents the number of faces
Or you'll need to memorize the properties for each Platonic solid--whichever you find easier:
- Tetrahedron: 4 vertices, 6 edges, 4 (equilateral) triangular faces
- Cube: 8 vertices, 12 edges, 6 square faces
- Octahedron: 6 vertices, 12 edges, 8 (equilateral) triangular faces
- Dodecahedron: 20 vertices, 30 edges, 12 pentagonal faces
- Icosahedron: 12 vertices, 30 edges, 20 (equilateral) triangular faces
Example 1: The number of Platonic solids is ___
- The answer is 5.
Example 2: Each face of an icosahedron has ___ sides
- Be careful here...you may be tempted to answer "20," but think of the face of the icosahedron (which is a triangle). The answer is 3.
Example 3: How many pentagons meet at each vertex of a Platonic dodecahedron?
- The answer is 3.
Example 4: A regular octahedron has ___ edges
- The answer is 12.
Example 5: How many different types of polygonal faces are used to form the Platonic solids?
- There are triangles (tetrahedron, octahedron, icosahedron), squares (cube), and pentagons (dodecahedron).
- The answer is 3.
Example 6: The sum of the number of faces, vertices, and edges of a Platonic hexahedron is ___
- A cube has 6 faces, 8 vertices, and 12 edges.
- 6 + 8 + 12 = 26.