10.1: Classifying Solid Figures
Introduction
Wrap it Up
Candice and Trevor have gotten a job working at the mall for the holiday season. They are both going to be working in the wrapping station. During the holidays, the mall offers free gift wrapping. People can come through and have their gifts wrapped. If they want to make a donation that money is used to help needy families.
Candice and Trevor both show up on their first day for training. Mrs. Scott, the manager of the wrapping station, shows them both where they will be working.
“First, we need to show you some great techniques for wrapping presents,” Mrs. Scott explains. “There are some ways that are more effective and useful than others.”
Candice and Trevor both take a seat in front of a bunch of different items.
There is a round bottle of perfume, a shoe box, a soccer ball, and a magician’s hat with a round bottom and a point at the top.
“What kinds of shapes to do you see here?” Mrs. Scott asks the two students.
Before seeing Candice and Trevor’s answers, think about this question yourself. Based on the descriptions, how would you classify these objects? Make a few notes in your notebook.
In this lesson, you will learn how to classify three-dimensional solids. By the end of this lesson, you will know how Candice and Trevor can classify each item.
What You Will Learn
In this lesson, you will learn to the following skills.
- Classify solid figures as prisms, cylinders, pyramids, cones, or spheres.
- Identify faces, edges, and vertices of solid figures.
- Describe and classify faces of solid figures as specific polygons.
- Identify patterns and describe relationships among the number of edges, vertices and faces of solid figures.
Teaching Time
I. Classify Solid Figures as Prisms, Cylinders, Pyramids, Cones or Spheres
In our earlier lessons, all of the figures that we have been working with have been plane figures which are two-dimensional figures. This means that they have only length and width. Even a circle which has a circumference and a diameter is still a plane figure. It does not have depth.
This lesson will focus on solid figures, which are three-dimensional figures.
How is a solid figure different from a plane figure?
A solid figure has length, width and height, whereas a plane figure only has length and width.
Let’s look at some plane figures.
Now let’s look at some solid figures.
We see solid figures around us every day. Take a look around you. What solids do you see? How many faces or edges do they have? Recognizing and understanding these figures is an important key to understanding geometry.
How can we identify different solid figures?
We can identify them according to the features that are unique to each type of solid.
A sphere is a solid figure that has no faces, edges, or vertices. This is because it is completely round; it has no flat sides or corners.
A cone has one face but no edges or vertices. Its face is in the shape of a circle. Because a circle is a flat, plane shape, it is a face. But because it is round around the outside, it does not form any edges or vertices. You will learn more about those in a minute.
A cylinder has two circular faces but also no edges or vertices.
Pyramids have one base and at least three triangular sides. A triangular pyramid has a triangular base and three other triangular faces, or four in all. A rectangular pyramid has a rectangular base and four other triangular faces, or five in all. It has edges and vertices where all of its faces meet, and always has a vertex at the top.
A prism is a solid figure that has two congruent parallel bases and some number of sides. In other words, it can have any number of faces, but at least two of them must be parallel. The shape of the two parallel bases can be a triangle, square, rectangle, pentagon, hexagon, or any other kind of polygon. The number of sides the bases have determines the number of faces the figure has. The number of edges and vertices also depends on the shape of the base. Here are some different prisms.
As you can see, there are many types of prisms. The shape of the bases tells us which kind of prism it is.
10A. Lesson Exercises
Identify each solid figure. Be as specific as you can.
Take a few minutes to check your answers with a partner.
II. Identify Faces, Edges and Vertices of Solid Figures
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 for classification.
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 the 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 five faces. One of the faces is a different shape from the others, since it is a square while the other faces are triangles. The different face 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 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.
III. Describe and Classify Faces of Solid Figures as Specific Polygons
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.
Let’s look at an example.
Example
First, you can see that each side is a polygon, and there are two matching and parallel sides. 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.
Let’s look at another one.
Example
Here we have a pyramid. You can tell that it is a pyramid because the lateral (side) faces all connect at one vertex. The base names this pyramid. A six sided polygon is a hexagon.
This is a hexagonal prism.
You can identify any prism or pyramid by examining the polygon that makes up the base. It isn’t necessary to worry about this with cones or cylinders. Remember that a cone has a circle for a base and so does a cylinder. The name of the solid is not changed because of the base.
IV. Identify Patterns and Describe Relationships among the Number of Edges, Vertices and Faces of Solid Figures
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 lateral (side) plus the base and top, or 5 in all. A hexagonal prism has 6 lateral faces plus the base and top, or 8 faces in all.
Example
A prism has a base with \begin{align*}n\end{align*}
A prism with \begin{align*}n\end{align*}
This means that we can put in any number for \begin{align*}n\end{align*}
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 be a nonagonal prism, and 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*}
Let’s apply this with a few examples.
Example
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, it is a septagonal prism.
Example
A figure has 10 faces, what is the base of the figure?
To work on this one, we have to work backwards. If the number of faces is \begin{align*}n + 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.
10B. Lesson Exercises
Name the number of faces in each.
- A base of a pentagonal prism
- A base of a nonagonal prism
- A base of a hexagonal prism
Check your work with a friend.
Now let’s use what we have learned to solve the problem from the introduction.
Real–Life Example Completed
Wrap it Up
Here is the original problem once again. Reread it and underline the three-dimensional solids that you need to identify.
Candice and Trevor have gotten a job working at the mall for the holiday season. They are both going to be working in the wrapping station. During the holidays, the mall offers free gift wrapping. People can come through and have their gifts wrapped. If they want to make a donation they can and that money is used to help needy families.
Candice and Trevor both show up on their first day for training. Mrs. Scott, the manager of the wrapping station, shows them both where they will be working.
“First, we need to show you some great techniques for wrapping presents,” Mrs. Scott explains. “There are some ways that are more effective and useful than others.”
Candice and Trevor both take a seat in front of a bunch of different items.
There is a round bottle of perfume, a shoe box, a soccer ball, and a magician’s hat with a round bottom and a point at the top.
“What kinds of shapes to do you see here?” Mrs. Scott asked the two.
As Trevor and Candice tell Mrs. Scott what types of figures are present on the table, let’s do our own inventory.
The round bottle of perfume is a cylinder.
The shoe box is a rectangular prism.
The soccer ball is a sphere.
The magician’s hat is in the shape of a cone.
How did you do? Go back and check the answers that you wrote at the beginning of the lesson. If you got them all correct, good work. If not, then make a note of which ones you mixed up to help yourself next time.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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 other faces meeting at one vertex.
- Edges
- the lines 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 an evenly rounded surface that ends with a point
- Cylinder
- a three-dimensional figure with two circular bases
- Vertex
- Where two or more edges meet
Technology Integration
Khan Academy Identifying Geometric Solids
- http://www.teachertube.com/members/viewVideo.php?video_id=185816&title=Prism__Cylinder__Sphere&vpkey= – This is a great video on simply identifying different solids based on their individual features.
Time to Practice
Directions: Identify each figure. Be as specific as possible.
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Directions: Count the number of faces, edges, and vertices in each figure.
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Directions: Answer each question.
16. A figure has one circular face, no edges, and no vertices. What kind of figure is it?
17. A figure has one pair of parallel sides that are circular. What kind of figure is it?
18. Decagons are polygons that have ten sides. How many faces does a decagonal prism have?
19. A hexagon has six sides. How many faces does a hexagonal prism have?
20. A heptagon has seven sides. How many faces does a heptagonal prism have?