Have you ever received a medal or trophy? Take a look at this dilemma.
“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 a 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 Concept, you will need to figure out the parts of a pentagonal prism.
Guidance
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. We 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.
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 a cylinder and a cone and a sphere.
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.
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.
If you know how to identify a figure, then you can draw it too. Take a look.
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.
Name each solid figure.
Example A
Solution: Rectangular Prism
Example B
Solution: Hexagonal Prism
Example C
Solution: Cone
Now let's go back to the dilemma from the beginning of the Concept.
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
- 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.
Guided Practice
Here is one for you to try on your own.
Sketch a cylinder.
Solution
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.
Video Review
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.