Have you ever thought about the parts of a solid figure?

Solid figures have their own characteristics, just like flat figures.

Candice and Trevor are wrapping gifts at the mall. A customer comes with a figure that has 10 faces. It is a unique type of jewelry box.

Do you know what the shape of the base of this figure is?

In this article, you will learn *how to identify a figure according to its faces, edges, and vertices*. At the end of this Section, you will discover the shape of the jeweler’s base.

## Faces, Edges, and Vertices of Solids

## Orientation

Previously, we worked on identifying solids. Well, now we need to know how to classify them more specifically.

To do this, let’s look at the characteristics of solid figures. The number of faces, edges, and vertices that a solid figure has tells us what type of solid figure it is. We can use this information to classify solids.

We classify or identify them by the number they have. Let’s start by looking at these faces.

A face is a flat side of a solid figure. Faces are shaped like flat shapes, such as triangles, rectangles, and squares. Check out the featured faces below.

Each solid figure has multiple faces. We can count the number of faces that the figure has. How many faces does each figure shown above have?

It has a face at the bottom and top base. It has four faces around the sides. Therefore it has six faces in total.

What shape do the faces have?

They are rectangles. We can name this figure a rectangular prism.

Yes. With prisms, you can use the shape you see to help you name what type of prism it is.

Now let’s look at another solid figure.

This figure only has one face. It is at the bottom and the sides are triangles that meet at a specific vertex. This is called a pyramid. Notice that the base of the pyramid is a square. This is called a square pyramid. The base gives the name to the figure.

Now that you understand about faces, let’s look at the edges. We can identify a solid figure by counting the edges.

An edge is a place where two faces meet. The edges are straight; they cannot be curved. How many edges does this figure have?

Count all the edges where two faces meet. This figure has 8 edges.

Some figures do not have edges because they do not have flat sides. Think of the cones, spheres, and cylinders. They do not have edges.

The place where two or more edges meet is called the vertex. A vertex is like a corner. We can count the number of vertices to identify solid figures.

This table can give you an idea of some of the faces, edges, or vertices of common solid shapes.

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

sphere |
0 | 0 | 0 |

cone |
one | 0 | 0 |

cylinder |
2 | 0 | 0 |

pyramid |
5 | 8 | 5 |

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

Sometimes you just have to count the faces, edges, and vertices to find out the number in each solid figure.

When we look at a rectangular prism, one of the first things we see is that the figure names the type of prism. This is especially true in prisms. When we look at a solid figure like a prism or a pyramid, we must think of polygons to find out what type of prism or pyramid the figure is.

In previous math classes, you may have named solid prisms or pyramids, but now you need to be more specific.

First, you can see that each side is a polygon. This means that we are working to identify a type of prism. Let’s use the base to identify the prism. The base is a five-sided figure. We know that a five-sided figure is called a pentagon.

**It is a pentagonal prism.**

When you think about the number of faces, vertices, and edges in solid shapes, you may notice that some patterns appear.

We can see a pattern of spheres, cones, and cylinders. Can you guess which one it is? To understand the pattern, we need to think about the number of faces, edges, and vertices that each shape has. All these figures are curved in a certain way, so they do not have edges or vertices. How about their faces? A sphere has no faces, a cone has a circular face, and a cylinder has two circular faces. Therefore, the number of faces increases in one from one figure to the other. This is a pattern.

**What about prisms? Is there a pattern here?**

There is definitely a pattern in relation to prisms. As the number of sides at the base and the cusp increases the parallel faces, the number of side faces increases the same amount.

A triangular prism, therefore, has 3 sides plus the lower and upper base, that is, 5 in total. A hexagonal prism has 6 sides plus the lower and upper base, that is to say, 8 faces in total.

A prism has a base with number n sides. How many faces does the prism have?

A prism with a number n from sides?

This means that we can insert any number to n. If we put 3 and make a triangular prism, how many faces will the prism have? As we said, it will have 3 side faces, a bottom, and a top base, or 5 faces.

What happens if we insert 6 for n and make a hexagonal prism? The figure will have 6 lateral faces plus the lower and upper base, that is to say, 8 faces in total. If we insert 9 for n, the figure will have 9 lateral faces, a lower base and an upper one, that is to say, 11 faces in total. Can you see the pattern?

In a prism, we always have a number of side faces that are determined by the number of sides of the polygon that is the base. So we add two because there is always a lower base and an upper one. In other words, to find the total number of faces we add 2 to the number of sides of the base. We can write a formula to help us understand this.

If the base has a number n sides then the prism will have a number n + 2 of faces.

**Here is another example.**

A base has seven sides. How many faces does it have?

If the base has 7 sides, we can use the formula to find the number of faces.

n + 2= number of faces

7 + 2 = 9

This figure has nine faces.

Now it’s your turn to practice. How many faces does each figure have, given the information about the shape of the base?

**Example A**

A base of a pentagon

Solution: 7 faces

**Example B**

A base of a nonagon

Solution: 11 faces

**Example C**

A base of a hexagon

Solution: 8 faces

Here is the original problem once again.

Solid figures have their own characteristics, just like flat figures.

Candice and Trevor are working wrapping gifts at the mall. A customer brings a figure that has 10 faces. It is a unique type of jewelry box.

Do you know what the shape of the base of the figure is?

To work on this exercise, we must go back. If the number of faces is n + 2, then the number of sides at the base will be x – 2.

10 is the number of faces. That’s ours.

10 – 2 = 8

The base is an 8-sided figure. An eight-sided figure is an octagon.

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**Vocabulary**

**Flat figure:**A two-dimensional figure.**Solid figure:**A three-dimensional figure.**Expensive:**The flat polygon of a solid figure. A figure can have more than one face.**Prism:**The flat polygon of a solid figure. A figure can have more than one face.**Pyramid:**A three-dimensional figure with a polygon as the base and all faces meet at a vertex.**Aristae:**The line where two faces meet.**It will be**A three-dimensional figure in which all points are equidistant from the center.**Cone:**A three-dimensional figure with a circular base and a side that meets at a vertex.**Cylinder:**A three-dimensional figure with two circular bases.**Vertex:**A point where two or more edges meet.