10.2: Faces, Edges, and Vertices of Solids
Have you ever thought about the parts of a solid figure?
Solid figures have their own characteristics, just like plane figures do.
Candice and Trevor are working at the wrapping paper station at the mall. A customer came in with a figure that has 10 faces. It was a unique kind of jewelry box.
Do you know what shape the base of the figure is?
In this Concept, you will learn how to identify a figure according to faces, edges and vertices. By the end of this Concept, you will know how to figure out the shape of the base of the jewelry box.
Guidance
Now that you know how to identify some solids, we need to look at how to classify them more specifically.
To do this, let’s look at the features of solid figures. The number of faces, edges, and vertices a solid figure has tells us what kind of solid figure it is. We can use this information to classify solids.
We classify, or identify, them by the number of each that they have. Let’s begin by looking at faces.
A face is a flat side of a solid figure. Faces are in the form of plane shapes, such as triangles, rectangles, and squares. Take a look at the faces highlighted below.
Every solid figure has several faces. We can count the number of faces the figure has. How many faces does each figure above have?
It has a face on the bottom and on the top. It has four faces around the sides. Therefore it has six faces in all.
What shape are the faces?
They are rectangles. We call this figure a rectangular prism.
Yes. With prisms, you can use the figure that you see to help you name the type of prism that it is.
Now let’s look at another solid.
This figure has only one face. It is on 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 of the pyramid is a square. This is called a square pyramid. The base names the figure.
Now that you understand faces, let’s look at edges. We can also identify a solid figure by counting the edges.
An edge is the place where two faces meet. Edges are straight; they cannot be curved. How many edges does this figure have?
Count all of the straight edges where two faces meet. This figure has 8 edges.
Some figures do not have any edges because they do not have flat sides. Think about cones, spheres and cylinders. They don’t have any edges.
The place where two or more edges meet is called a vertex. A vertex is like a corner. We can also count the number of vertices in order to identify solid figures.
This chart can give you an idea of some of the faces, edges of vertices of common solid figures.
Figure Name | Number of Faces | Number of Edges | Number of Vertices |
---|---|---|---|
sphere | 0 | 0 | 0 |
cone | 1 | 0 | 0 |
cylinder | 2 | 0 | 0 |
pyramid | 5 | 8 | 5 |
prism | at least 5 | at least 9 | at least 6 |
Sometimes, you will just have to count the faces, edges and vertices to figure out the number that are in each solid figure.
When we looked at the rectangular prism, one of the first things that you can see is that the figure names the type of prism. This is especially true or prisms. When we look at a solid figure such as a prism or a pyramid, we have to use our thinking about polygons to figure out which type of prism or pyramid the figure is.
In earlier math classes, you might have just called the solid a prism or a pyramid, but now you need to be more specific.
First, you can see that each side is a polygon. That means that we are working on identifying a type of prism. Let’s use the base to help identify this prism. The base is a five sided figure. We know that a five sided figure is called a pentagon.
This is a pentagonal prism.
When you think about the number of faces, vertices and edges in solid figures, you may start to see some patterns emerge.
We can see one pattern in spheres, cones, and cylinders. Can you guess what it is? To understand the pattern, we need to think about the number of faces, edges, and vertices each figure has. All of these figures are curved in some way, so they have no edges or vertices. What about their faces? A sphere has no faces, a cone has one circular face, and a cylinder has two circular faces. Therefore the number of faces increases by one from one figure to the next. This is a pattern.
What about prisms? Is there a pattern there?
There is definitely a pattern with regard to prisms. As the number of sides in the base and top parallel faces increases, the number of side faces increases the same amount.
A triangular prism therefore has 3 sides plus the base and top, or 5 in all. A hexagonal prism has 6 sides faces plus the base and top, or 8 faces in all.
A prism has a base with \begin{align*}n\end{align*} number of sides. How many faces does the prism have?
A prism with \begin{align*}n\end{align*} number of sides?
This means that we can put in any number for \begin{align*}n\end{align*}. If we put in 3 and make this a triangular prism, how many faces will the prism have? As we said, it will have 3 side faces, a top, and a base, or 5 faces.
What if we put 6 in for n and make it a hexagonal prism? The figure will have 6 side faces plus the base and top, or 8 faces in all. If we put 9 in for n, the figure would have 9 side faces, a top, and a base, or 11 faces in all. Do you see the pattern?
In a prism, we always have a number of side faces determined by the number of sides in the polygon that is the base. Then we add two, because there is always a base and top. In other words, to find the total number of faces we add 2 to the number of the base’s sides. We can write a formula to help us to understand this.
If the base has \begin{align*}n\end{align*} number of sides, then the prism will have \begin{align*}n + 2\end{align*} number of faces.
Here is another one.
A base has seven sides. How many faces does it have?
If the base has 7 sides, then we can use the formula to find the number of faces.
\begin{align*}n+2 & = number \ of \ faces\\ 7+2 & = 9\end{align*}
This figure has nine faces.
Now it's time for you to practice. How many faces does each figure have given the provided 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 plane figures do.
Candice and Trevor are working at the wrapping paper station at the mall. A customer came in with a figure that has 10 faces. It was a unique kind of jewelry box.
Do you know what shape the base of the figure is?
To work on this one, we have to work backwards. If the number of faces is \begin{align*}n + 2\end{align*}, then the number of sides in the base would be \begin{align*}x - 2\end{align*}.
10 is the number of faces-that is our \begin{align*}x\end{align*}.
\begin{align*}10 - 2 = 8\end{align*}
The base is an 8 sided figure. An eight sided figure is an octagon.
Vocabulary
Here are the vocabulary words in this Concept.
- Plane figure
- a two-dimensional figure
- Solid figure
- a three-dimensional figure
- Face
- the flat polygon of a solid figure. A figure can have more than one face.
- Prism
- a three-dimensional figure with two parallel congruent polygons as bases
- Pyramid
- a three-dimensional figure with one polygon for a base and all faces meet at one vertex.
- Edges
- the line where two faces meet
- Sphere
- a three-dimensional figure where all points are equidistant from one center point.
- Cone
- a three-dimensional figure with a circular base and one side that meets at one vertex
- Cylinder
- a three-dimensional figure with two circular bases
- Vertex
- Where two or more edges meet
Guided Practice
Here is one for you to try on your own.
Name the figure and the number of faces.
Answer
Here we have a pyramid. You can tell that it is a pyramid because the faces all connect at one vertex. The base names this pyramid. You can see that it has six sides to it. A six sided polygon is a hexagon.
This is a hexagonal prism.
The base is a hexagon.
\begin{align*}n + 2\end{align*} = number of faces.
The base of a hexagon has six sides.
\begin{align*}6 + 2 = 8\end{align*}
This figure has eight faces.
Video Review
Here is a video for review.
- This is a video on vertices and diagonals of solid figures.
Practice
Directions: Count the number of faces, edges, and vertices in each figure.
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Directions: Answer each question.
13. A figure has one circular face, no edges, and no vertices. What kind of figure is it?
14. A figure has one pair of parallel sides that are circular. What kind of figure is it?
15. Decagons are polygons that have ten sides. How many faces does a decagonal prism have?
16. A hexagon has six sides. How many faces does a hexagonal prism have?
17. A heptagon has seven sides. How many faces does a heptagonal prism have?
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