Mack is building a garden for his mother. He wants to put in pentagonal stones to decorate the area. In order to make the best arrangement, he needs to find the number of faces and edges in a pentagonal prism. How many faces and edges are there in a pentagonal prism?

In this concept, you will learn how to classify solid figures by identifying faces, edges and vertices.

### Solid Figures

Solid figures are shapes that exist in three-dimensional space. Unlike plane figures, which have only length and width, solid figures have length, width, and height. Let’s take a look at identifying solid figures.

Here are three different **prisms**. Each figure has two common congruent bases and these bases are hexagons, pentagons and triangles. The sides of each figure are made up of rectangles. You name these figures according to the base: a hexagonal prism, a pentagonal prism and a triangular prism. Notice that the key with prisms is that the sides are rectangles.

Another type of solid figure is called a **pyramid**. A pyramid has a base and triangular sides that meet at a single vertex. You identify a pyramid according to its base.

Here are some pyramids.

There are other solid figures too that have circles in them. See below a cylinder, a cone, and a sphere.

Solid figures have **faces**, **edges** and **vertices**. You can use the number of faces, edges and vertices to identify the solid figures.

A **face** is the flat side of a solid figure. Faces are in the form of plane shapes, such as triangles, rectangles, and pentagons. An **edge** is the place where two faces meet. Edges are straight; they cannot be curved. Vertices or a **vertex** is the point where edges meet.

You can identify the three parts of a solid by looking at the following diagram.

Once you know how to identify the faces, edges and vertices of a solid, you can count them too.

Let’s look at an example.

How many faces, edges, and vertices does the figure below have?

First, let’s count the faces. Remember, each face is a flat plane shape. In this figure, the bases, or top and bottom, are hexagons and the sides are all rectangles. There are six faces around the sides and two bases. This figure has eight faces in all.

Next, let’s count the edges where each face meets another. There are six around the top hexagon where it meets each side, and six more around the bottom hexagon where it meets each side. And there are six more where each side meets another. This figure has 18 edges.

Then, let’s find the vertices. Remember, a vertex is like a corner. This figure has six corners, or vertices, on the top and six on the bottom. This figure has twelve vertices in all.

You can count faces, edges and vertices of all the solids. This information can also help us to classify them. If you think about prisms and pyramids, you can think about the number of edges, faces and vertices.

However, if you think about a sphere, a cone and a cylinder, you will notice that faces, edges and vertices don’t apply to all of these.

Let’s look at a chart to help us classify solid figures according to their faces, edges and vertices.

Figure Name |
Number of Faces |
Number of Edges |
Number of Vertices |

sphere | 0 | 0 | 0 |

cone | 1 | 0 | 0 |

cylinder | 2 | 0 | 0 |

triangle pyramid | 4 | 6 | 4 |

square pyramid | 5 | 8 | 5 |

prism | at least 5 | at least 9 | at least 6 |

Let’s look at an example.

Draw a triangular prism.

First, you know that a triangular prism has two bases shaped like triangles. To draw the triangular prism, let’s begin by drawing its base.

Next, let’s draw the side that is facing toward the front. You know that a prism has rectangular sides that are all the same height. You also know that the bottom edge of the rectangular side connects to one edge of the base, so you can draw the rectangular face attached to the base.

Notice that the top face is exactly the same size and shape as the base, only it is connected to a top edge of the rectangular side. Imagine you could slide the base triangle up and put it on top of the rectangle.

It becomes easy to figure out how to draw this figure once you understand that the bases are triangles and as with any prism, the sides are rectangles. Connecting them together forms the solid figure.

### Examples

#### Example 1

Earlier, you were given a problem about Mack’s stone pattern.

Mack needs to know the number of faces and edges on a pentagonal prism in order to build a pattern of stone in the garden.

First, you can use the fact that

gives us the pattern for the number of faces in a prism. In this pattern, represents the number of sides in the base. The pentagon has five sides, so we know that is 5.\begin{align*}\begin{array}{rcl}
\text{Faces} &=& n + 2 \\
\text{Faces} &=& 7 \ \text{faces}
\end{array}\end{align*}

Next, draw a picture of a pentagonal prism.

From the drawing, you can count the edges.

Faces: 7

Edges: 15

Vertices: 10

The answer is 7 faces, 15 edges, and 10 vertices.

#### Example 2

Sketch a cylinder.

First, remember that cylinders have two circular bases. Therefore you will need to draw a circular base which is the same as the circular top face.

Next, add the front view. Cylinders do not have a side face. They are curved. Imagine holding up a cylinder and looking at it from the side. What would it look like? From the side, the cylinder would appear to have a rectangular face. This is a bit of an illusion, but you should sketch the cylinder as you would see it. Even though the side meets the base around the curve of the circle, you can draw a rectangle. But let’s show the top and bottom of the rectangular side with dashed lines so that you know there isn’t really a straight edge there.

There aren’t any vertices to connect between the top and bottom faces since they are round, so you’re done.

#### Example 3

Name the figure below and name the number of vertices, faces, and edges.

First, name the figure.

Rectangular Prism

Next, find the number of vertices, faces, and edges.

Vertices: 8

Faces: 6

Edges: 12

The answer is there are 8 vertices, 6 faces, and 12 edges.

#### Example 4

Name the figure below and name the number of vertices, faces, and edges.

First, name the figure.

Hexagonal Prism

Next, find the number of vertices, faces, and edges.

Vertices: 12

Faces: 8

Edges: 18

The answer is there are 12 vertices, 8 faces, and 18 edges.

#### Example 5

Name the figure below and name the number of vertices, faces, and edges.

First, name the figure.

Cone

Next, find the number of vertices, faces, and edges.

Vertices: 0

Faces: 1

Edges: 0

The answer is there are 0 vertices, 1 faces, and 0 edges.

### Review

Answer the following questions about each solid figure.

1. What is the name of this figure?

2. How many faces does it have?

3. How many vertices does it have?

4. How many edges does it have?

5. What is the name of this figure?

6. How many faces does it have?

7. How many edges does it have?

8. How many vertices does it have?

9. What is the name of this figure?

10. How many faces, edges and vertices does it have?

11. What is the name of this figure?

12. How many faces does it have?

13. How many edges does it have?

14. How many vertices does it have?

15. A figure has one circular face, no edges, and no vertices. What kind of figure is it?

16. A figure has one pair of parallel sides that are circular. What kind of figure is it?

17. Sketch a cone.

18. Sketch a pentagonal prism.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.5.