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Faces, Edges, and Vertices of Solids

Leaning on what makes a solid, we identify the elements, including faced, edges, and vertices, in the following lesson. Learn more.

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Faces, Edges, and Vertices of Solids
License: CC BY-NC 3.0

Martha is visiting the famous Louvre, in Paris, France.  She is awed by the fascinating structure outside the museum.  She wants to identify the structure according to its attributes. 

What is the name of this solid figure and how many faces does it have?

In this concept, you will learn how to identify a figure according to faces, edges and vertices, as well as by the shape of the base.

Using Faces, Edges, and Vertices to Identify a Solid

Every solid figure, with the exception of a sphere, has one or more faces, or flat surfaces.

Let's look at an example.

The figure above has a face on the bottom, called the base, and another on the top. It has four faces around the sides. Therefore, it has six faces in all.

The shapes of the faces are rectangles. Since the shape on the bottom and the top are the same, the general category of this shape is called a prism.  Prisms are more specifically named by the shape of the base.  In this case, the shape of the base is a rectangle; so this solid figure is called a rectangular prism.

Let's look at an example.

License: CC BY-NC 3.0

This figure has four faces. The base, or the face on the bottom, is a triangle, and the sides are triangles that meet at a vertex at the top. The general category for this solid figure is a pyramid. Notice that the base of the pyramid is a triangle. The specific name of this soid figure is called a triangular pyramid. As with the prism, the base indicates the specific name of the figure.

You 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.

Let's look at an example.

License: CC BY-NC 3.0

Count all of the edges on the solid figure above. This figure has 9 edges.

Cones, spheres, and cylinders do not have any edges because they do not have any flat sides.

The place where two or more edges meet is called a vertex. A vertex is like a corner. If you have more than one vertex they are called vertices.  You can also count the number of vertices in order to identify solid figures.

Let's look at an example.

License: CC BY-NC 3.0

When you count the number of vertices in the figure above, you see there are 7 vertices.  All solid figures that have a vertex at the top are classified as pyramids.

This chart gives 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 at least 4 at least 6 at least 4
prism at least 5 at least 9 at least 6

A prism has the same polygonal shape on the top as it has on its base.  A pyramid has a polygonal shape as its base, and a vertex at the top where all of its triangular-shaped side faces meet.

Let's look at an example.

In the figure above, you can see that the base and the top of the figure are the same polygonal shape. That means that you are working on identifying a type of prism. Let’s use the base to help identify the specific type of prism. The base is a five-sided figure. You know that a five-sided figure is called a pentagon.  This is a pentagonal prism.

Let's look at an example.

License: CC BY-NC 3.0

In the figure above, the base is a polygon and the top forms a vertex.  This means the figure is a pyramid.  The base is a six-sided figure.  You know that a six-sided figure is called a hexagon.  This solid figure is a hexagonal pyramid.

When you think about the number of faces, vertices and edges in solid figures, you may start to see patterns emerge.

You can see a pattern related to spheres, cones, and cylinders. 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?  As the number of edges of the base (and parallel top) increases, the number of side faces increases the same amount.  The base of a triangular prism has 3 edges.  It therefore has 3 side faces plus the base and top, or 5 in all. A hexagonal prism has 6 side faces plus the base and top, or 8 faces in all.

This means that you can put in any number for \begin{align*}n\end{align*}, representing the number of edges of the base (and top).  You know this also represents the number of faces that make up the sides.  You then add the base as a face and the top as another face, adding two more faces.  Therefore, if the base has \begin{align*}n\end{align*} number of edges, then the prism will have \begin{align*}n + 2\end{align*} number of faces.

Now, let's consider pyramids and the pattern that exists here.  As with the prism, the number of edges of the base will equal the number of faces of the triangular sides.  But there is not a top face to add to this since the top is the vertex where the triangular sides meet.  For example, the base of a triangular pyramid has 3 edges.  As with the triangular prism, it therefore has three side faces.  In the case of pyramid, these side faces are triangular with the faces meeting to form the vertex.  You then add the three side faces with the base face to get four faces in total.

This means that you can put in any number \begin{align*}n\end{align*}, representing the number of edges of the base.  You know this also represents the number of triangular faces that make up the sides.  You then add the base as a face, adding one more face. Therefore, if the base has \begin{align*}n\end{align*} number of edges, then the pyramid will have \begin{align*}n + 1\end{align*} number of faces.

Examples

Example 1

Earlier, you were given a problem about Martha and the Louvre.

She notes that the base of the structure has 4 edges.  Using this information, what is the name of this figure and how many faces does it have?

First, you know that the number of edges of the base is the same number as that of the side faces.  There are 4 side faces.

Next, see that there is a vertex at the top and notice that the side faces are triangles.  You know this is a pyramid.

Then, use the formula to calculate the number of faces of a pyramid (\begin{align*}n + 1\end{align*}):  4 + 1 = 5

The answer is that the Louvre Pyramid is a square pyramid with 5 faces.

Example 2

Name the figure and the number of faces.

First, observe that the side faces of the figure are all triangular in shape forming a vertex at the top.  You know this is a pyramid.

Next, see that the base is a hexagon with 6 edges.

Then, using the formula  \begin{align*}n + 1\end{align*}to calculate the number of faces of a pyramid , calculate the number of faces in this figure as 7 in total.

The answer is a hexagonal pyramid with 7 faces.

Here is another one:

A figure has a base and parallel top face, each with 7 edges. How many faces does it have?

First, if the base has 7 edges, there must be 7 side faces.  

Next, since there is a parallel top face, you know this is a prism.

Then, you use the formula for calculating the number of faces in a prism:

\begin{align*}n+2 & = \mathrm{number \ of \ faces}\\ 7+2 & = 9\end{align*}

The answer is that the prism has 9 faces.

Name the figure and the number of faces.

Example 3

License: CC BY-NC 3.0

First, see that the base and the top are parallel, each with 9 edges.  Since there is a base with a parallel top, you know this is a prism.  You also know that a 9-sided figure is a nonagon.

Next, you know that if the base has 9 edges, there are 9 side faces.

Then, you use the formula for calculating the number of faces of a prism:  

\begin{align*}n + 2&= \mathrm{ number \ of \ faces}\\ 9 + 2&= 11\end{align*}The answer is this solid figure is a nonagonal prism with 11 faces. 

Example 4

License: CC BY-NC 3.0

First, note that the side faces of the figure are all triangular, creating a vertex at the top.  You know this is a pyramid.

Next, observe that the base is a septagon with 7 edges.

Then, use the formula to calculate the number of faces in the figure.

\begin{align*}n+1&= \text {number of faces in a pyramid}\\ 7 + 1 &= 8 \ \mathrm{ faces} \end{align*}

The answer is the figure is a septagonal pyramid with 8 faces.

Example 5

A prism with a base of a pentagon

First, you know that the base is a pentagon with 5 edges.

Next, you know that the prism has a base with a parallel top.

Then, calculate the number of faces using the formula for a prism (\begin{align*}n + 2\end{align*}):

5 + 2 = 7

The answer is this solid figure is a pentagonal prism with 7 faces.

Review

Count the number of faces, edges, and vertices in each figure.

  1.  

Answer each question.

  1. A figure has one circular face, no edges, and no vertices. What kind of figure is it?
  2. A figure has one pair of parallel sides that are circular. What kind of figure is it?
  3. Decagons are polygons that have ten sides. How many faces does a decagonal prism have?
  4. A hexagon has six sides. How many faces does a hexagonal prism have?
  5. A heptagon has seven sides. How many faces does a heptagonal prism have?

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.2.

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Vocabulary

Edge

An edge is the intersection between two faces of a figure. An edge is a line segment.

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