Faces of Polyhedra

Section 1.1 Faces of Polyhedra

Our first task is to identify the shapes of all the faces of a polyhedron. We need to know what the shapes are and how they are connected to each other.

🔗

Example 1.1.1.

Start by rotating this image so you have seen all the faces (there is a bottom).

🔗

Four of the faces are triangles. The fifth face is a square. Thus there are a total of 5 faces.

🔗

Each triangle shares two sides with other triangles and the third side with the square.

🔗

Figure 1.1.2. Crystal Form

🔗

Checkpoint 1.1.3.

Start by rotating this image so you have seen all the faces.

🔗

(a)

List the type of shape for each face. For example, triangles, pentagons, and squares.

🔗

(b)

For each shape, state how many faces have that shape. For example, 5 triangles, 1 pentagon, 5 squares.

🔗

(c)

Describe how each face is connected to other faces. For example, each triangle is adjacent to 2 triangles and one square.

🔗

Figure 1.1.4. Crystal Form

🔗

In this next example we describe faces that are a little less regular.

🔗

Example 1.1.5.

Start by rotating this image so you have seen all the faces.

🔗

The top and bottom faces are regular hexagons. Four of the faces are pentagons that are not regular. The other four faces have four sides (trapezoids). Thus there are a total of 10 faces.

🔗

Each hexagon is adjacent to the four pentagons and two of the trapezoids. Each pentagon is next to another pentagon and two of the trapezoids. Each trapezoid is next to another trapezoid and has pentagons on opposite sides. The pairs of trapezoids are adjacent to opposite sides of the hexagons. The two pairs of pentagons are likewise adjacent to opposite sides of the hexagons.

🔗

Figure 1.1.6. A Crystal Form

🔗

Checkpoint 1.1.7.

Start by rotating this image so you have seen all the faces.

🔗

(a)

List the type of shape for each face. For example, triangles, pentagons, and squares.

🔗

(b)

For each shape, state how many faces have that shape. For example, 5 triangles, 1 pentagon, 5 squares.

🔗

(c)

Describe how each face is connected to other faces. For example, each triangle is adjacent to 2 triangles and one square.

🔗

Figure 1.1.8. A Crystal Form

🔗

Prev Top Next