8.3: Solid Figures
Introduction
The Awards
“Rather than medals, I think it would be really cool if we used a pyramid as an award,” Jose said at lunch on Monday.
“I don’t. A medal is what is traditional and I think we should stick with medals,” Travis disagreed.
“Well, medals are so flat. I like the idea of being different. Besides, this is our first Olympics we could make our award interesting and unique,” Marcy added in.
“What about a prism instead? Something cool like pentagonal prism. I’ll draw my idea out and then we can take a look at it,” Jose said taking out a piece of paper and pencil.
“What is that?” Marcy asked leaning over his shoulder.
“A pentagonal prism,” Jose said.
“Nope, you have the faces and edges all wrong.”
How many faces and edges are there in a pentagonal prism? When working with solid figures, there are patterns that can help you to figure out this information. At the end of this lesson you will need to figure out the parts of a pentagonal prism.
What You Will Learn
In this lesson you will learn how to do the following skills.
- Classify solid figures.
- Identify and count faces, edges and vertices of solid figures.
- Sketch representations of solid figures as perspective or top – front – side views.
- Identify patterns and describe relationships among the number of edges, vertices and faces of solid figures.
Teaching Time
I. Classify Solid Figures
In this lesson we will examine solid figures. Solid figures are shapes that exist in three-dimensional space. Unlike plane shapes, which have only length and width, solid figures have length, width, and height. We will learn to identify solid figures and their properties. Let’s take identifying solid figures.
Here are three different prisms. Notice that there are two common congruent bases-hexagons, pentagons and triangles here, and the sides are made up of rectangles. We call these 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. We identify a pyramid according to its base. Here are some pyramids.
There are other solid figures too that have circles in them. Here is an example of a cylinder and a cone and a sphere.
II. Identify and Count Faces, Edges and Vertices of Solid Figures
Solid figures have faces, edges and vertices. We can use the number of faces, edges and vertices to identify the solid figures.
What is a face?
A face is the flat side of a solid figure. Faces are in the form of plane shapes, such as triangles, rectangles, and pentagons.
What is an edge?
An edge is the place where two faces meet. Edges are straight; they cannot be curved.
What is a vertex?
Vertices or a vertex is the point where edges meet. We often think of them as the points of a figure.
We 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.
Example
How many faces, edges, and vertices does the figure below have?
Let’s count the faces first. 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.
Now 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. It has twelve vertices in all.
We 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 theses. 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 |
triangular pyramid | 4 | 6 | 4 |
square pyramid | 5 | 8 | 5 |
prism | at least 5 | at least 9 | at least 6 |
Copy this chart down in your notebook.
III. Sketch Representations of Solid Figures as Perspective or Top-Front-Side Views
Now that we understand faces and how they fit together, let’s try combining them to draw solid figures. Now instead of analyzing their components, we will be putting the pieces together. Let’s see how this works with a few examples.
Example
Draw a triangular prism.
We 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. We know that a prism has rectangular sides that are all the same height. We also know that the bottom edge of the rectangular side connects to one edge of the base, so we can draw the rectangular face attached to the base.
Now we have shown the width and height of the prism. Let’s draw the top face next. 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.
We now have shown the front and top views of the prism. All we need to do is connect any other vertices in the top face with the corresponding vertices in the base. In this case, we only need to draw one more edge connecting the top and bottom triangles
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.
Example
Sketch a cylinder.
Again, let’s think about the bases first. Cylinders have two circular bases. Therefore we’ll need to draw a circular base and a circular top face. Let’s draw the base first.
Next, we add the front view, as we did when we drew the rectangular side of the prism above. Cylinders, however, 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 we should sketch the cylinder as we would see it. Even though the side meets the base around the curve of the circle, we can draw a rectangle. But let’s show the top and bottom of the rectangular side with dashed lines so that we know there isn’t really a straight edge there.
Now we can add the top face. Remember, in a cylinder, the top and bottom face are exactly the same. Imagine you could slide the base up to the top of the rectangle and draw it again. This gives us the top face. There aren’t any vertices to connect between the top and bottom faces since they are round, so we’re done. We have drawn a cylinder that looks like this.
Drawing solids is not easy and may take a bit of practice. Be sure you understand what all of the faces, edges, and vertices of the figure look like.
Here are the steps:
Begin with the base, then draw the front (which is a side face), then the top, and, finally, connect the remaining vertices between the top and the bottom. Curved figures such as cylinders and cones can be especially tricky. From the side, a cylinder looks like it has a rectangular side face, and a cone looks like it has a triangular side face. Hold up some cylindrical and conical objects to see for yourself.
Write these steps down in your notebook.
IV. Identify Patterns and Describe Relationships among the Number of Edges, Vertices, and Faces of Solid Figures
Now that we can classify and draw solid figures, let’s analyze them in another way. By examining solid figures, we can find patterns and relationships among them. The patterns and relationships, in turn, help us to understand the similarities and differences among solid figures.
We can see one pattern in spheres, cones, and cylinders. 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 1 from one figure to the next. This is a pattern.
Another important relationship to recognize in solid figures is parallelism. We can identify prisms, for example, by their pair of parallel faces. In fact, we can say that opposite sides are always parallel.
Let’s look at a few examples where we can see these patterns.
Example
Identify the parallel faces in each solid figure.
How many pairs of parallel faces does the first figure have? Well, two of the faces are triangles, and three are rectangles. The triangles are the base and top, so they are parallel. Are any of the sides opposite each other? They are not.
The next figure, however, has more than one pair of parallel sides. In fact, it has three pairs. Can you find them all? Its base and top are parallel rectangles. The front and back faces are also parallel because they are opposite each other. The smaller side faces are opposite each other, so these, too, are parallel. We can see these pairs of opposites if we look at the rectangular base alone. A rectangle has two pairs of parallel sides. A rectangular prism has faces along those sides, so the faces must be parallel. Then we have one more pair for the third dimension of height (the base and top).
We can conclude that prisms whose base has an even number of sides have more than one pair of opposite parallel sides. Prisms whose base has an odd number of sides will only have a parallel relationship between the top and bottom, not any of the sides.
We can also understand one more relationship in prisms from this example. As the number of sides in the base and top 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.
Example
A prism has a base with \begin{align*}n\end{align*} number of sides. How many faces does the prism have?
A base with \begin{align*}n\end{align*} number of sides? This means that we can put in any number for \begin{align*}n\end{align*}. This is where patterns can be used in problem solving.
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 \begin{align*}n\end{align*} 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 \begin{align*}n\end{align*}, 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. If the base has n number of sides, then the prism will have \begin{align*}n + 2\end{align*} number of faces.
Real-Life Example Completed
The Awards
Here is the original problem once again. Reread it and then figure out the faces, edges and vertices of a pentagonal prism. Then draw a picture of one.
“Rather than medals, I think it would be really cool if we used a pyramid as an award,” Jose said at lunch on Monday.
“I don’t. A medal is what is traditional and I think we should stick with medals,” Travis disagreed.
“Well, medals are so flat. I like the idea of being different. Besides, this is our first Olympics we could make our award interesting and unique,” Marcy added in.
“What about a prism instead? Something cool like pentagonal prism. I’ll draw my idea out and then we can take a look at it,” Jose said taking out a piece of paper and pencil.
“What is that?” Marcy asked leaning over his shoulder.
“A pentagonal prism,” Jose said.
“Nope, you have the faces and edges all wrong.”
Remember, there are two parts to your answer.
Solution to Real – Life Example
To figure out the faces, edges and vertices of a pentagonal prism, we can look at patterns. First, we know that \begin{align*}n + 2\end{align*} gives us the pattern for the number of faces in a prism. In this pattern, \begin{align*}n\end{align*} represents the number of sides in the base. The pentagon has five sides, so we know that \begin{align*}n\end{align*} is 5.
\begin{align*}Faces &= n + 2\\ Faces &= 7 \ faces\end{align*}
Now we can draw a picture of a pentagonal prism.
From the drawing, you can count the edges. There are 15 edges in the pentagonal prism.
There are 10 vertices in the pentagonal prism.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Solid Figures
- three-dimensional figures with length, width and height.
- Prisms
- three-dimensional figures with polygons as bases and rectangles for side faces.
- Pyramid
- three-dimensional figures with a polygon as a base and side triangular faces that meet in a single vertex.
- Face
- the flat surfaces of a three-dimensional figure.
- Edge
- the place where two line segments meet in a three-dimensional figure.
- Vertex
- the point where edges meet in a three-dimensional figure.
Time to Practice
Directions: Answer the following questions about each solid figure.
- What is the name of this figure?
- How many faces does it have?
- How many vertices does it have?
- How many edges does it have?
- What is the name of this figure?
- How many faces does it have?
- How many edges does it have?
- How many vertices does it have?
- What is the name of this figure?
- How many faces, edges and vertices does it have?
- What is the name of this figure?
- How many faces does it have?
- How many edges does it have?
- How many vertices does it have?
- A figure has one circular face, no edges, and no vertices. What kind of figure is it?
- A figure has one pair of parallel sides that are circular. What kind of figure is it?
- Sketch a cone.
- Sketch a pentagonal prism.
- How many pairs of parallel faces does an octagonal prism have?
- How many pairs of parallel faces does a pentagonal prism have?
- Dodecagons are polygons that have twelve sides. How many faces does a dodecagonal prism have?
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