**High School Number Sense Lesson 46: Polyhedrons**

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**):

**4 faces: Tetrahedron**

6 faces: Hexahedron (cube)

8 faces: Octahedron

12 faces: Dodecahedron

20 faces: Icosahedron

6 faces: Hexahedron (cube)

8 faces: Octahedron

12 faces: Dodecahedron

20 faces: Icosahedron

**Platonic solids**. Number sense tests may ask for the number or shape of the faces, vertices (points), or edges (connecting the vertices). Or they may ask you for some combination of these.

You'll need to know

**Euler's Formula**, which is:

**V - E + F = 2**

**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**.

**Here's a free worksheet to help you practice Polyhedron:**

polyhedron.pdf |

**Up Next for High School: BaseAdd**