7.2: Area of Nets; Nets to Prisms
Learning Objectives
- Identify and understand the properties of polyhedra.
- Identify regular Platonic polyhedra.
- Identify, draw, and construct nets for solids.
Introduction
You have already learned that a polygon is a 2-dimensional figure that is made of three or more points joined together by line segments. Examples of polygons include triangles, quadrilaterals, pentagons, or octagons.
In general, an \begin{align*}n-\end{align*}gon is a polygon with \begin{align*}n\end{align*} sides. A triangle is a 3-gon or 3-sided polygon, and a pentagon is a 5-gon or 5-sided polygon. Below are examples of polygons:
You can use polygons to construct a 3-dimensional figure called a polyhedron. The plural of polyhedron is polyhedra.
A polyhedron is a 3-dimensional figure that is made up of polygon faces. A face is an outer side (or boundary) of a polyhedron. A cube is an example of a polyhedron, and its faces are squares (quadrilaterals).
Reading Check:
1. Fill in the blank:
A 3-dimensional figure made of polygon faces is called a _______________________.
2. Fill in the blank:
The plural form of polyhedron is _____________________________.
3. What shape is an example of a polyhedron?
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Polyhedron or Not
A polyhedron has the following properties:
- It is a 3-dimensional figure.
- It is made of polygons and only polygons. Each polygon is called a face of the polyhedron.
- Polygon faces join together along segments called edges.
- Each edge joins exactly two faces.
- Edges meet in points called vertices (plural form of vertex).
- There are no gaps between edges or vertices.
To review:
A polyhedron is a ____-dimensional shape that is made up of ____________________ called faces of the polyhedron. The faces are joined together at ________________ and the edges meet at _______________________.
Example 1
Is the figure a polyhedron?
Yes. A figure is a polyhedron if it has all of the properties of a polyhedron. This figure:
- Is 3-dimensional.
- Is constructed entirely of flat polygons (triangles and rectangles).
- Has faces that meet in edges and edges that meet in vertices.
- Has no gaps between edges.
- Has no non-polygon faces (like curves).
- Has no concave faces.
Since the figure has all of the properties of a polyhedron, it is a polyhedron.
Example 2
Is the figure a polyhedron?
No. This figure has faces, edges, and vertices, but all of its surfaces are not flat polygons. Look at the end surface marked \begin{align*}A\end{align*}. It is flat, but it has a curved edge so it is not a polygon. Surface \begin{align*}B\end{align*} is not flat (or planar).
Example 3
Is the figure a polyhedron?
No. The figure is made up of polygons and it has faces, edges, and vertices.
But the faces do not fit together — the figure has gaps. The figure also has an overlap that creates a concave surface. For these reasons, the figure is not a polyhedron.
Reading Check:
1. If a figure has curved edges, is it a polyhedron? __________
2. If a figure has gaps or overlapping faces, is it a polyhedron? __________
3. List 3 characteristics of polyhedra:
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Face, Vertex, Edge, Base
As you have learned, a polyhedron joins faces together along edges, and edges together at vertices. The following statements are true of any polyhedron:
- Each edge joins exactly two faces.
- Each edge connects exactly two vertices.
Again,
Each edge joins two _____________________.
Each edge connects two ___________________________.
To see why this is true, take a look at this prism. Each of its edges joins two faces along a single line segment. Each of its edges connects exactly two vertices.
Let’s count the number of faces, edges, and vertices in a few typical polyhedra.
The figures below are square pyramids. The square pyramid gets its name from its base, which is a square. It has 5 faces, 8 edges, and 5 vertices:
Other figures have a different number of faces, edges, and vertices:
If we make a table that summarizes the data from each of the figures we get:
Figure | Vertices | Faces | Edges |
---|---|---|---|
Square pyramid | 5 | 5 | 8 |
Rectangular prism | 8 | 6 | 12 |
Octahedron | 6 | 8 | 12 |
Pentagonal prism | 10 | 7 | 15 |
Calculate the sum of the number of vertices and edges. Then compare that sum to the number of edges. Fill in the empty boxes with the numbers from above!
Figure |
Vertices \begin{align*}v\end{align*} |
Faces \begin{align*}f\end{align*} |
Edges \begin{align*}e\end{align*} |
Vertices + Faces \begin{align*}v + f\end{align*} |
---|---|---|---|---|
Square pyramid | 5 | 8 | \begin{align*}5 + 5 = 10\end{align*} | |
Rectangular prism | 6 | 12 | \begin{align*}8 + 6 = 14\end{align*} | |
Octahedron | 6 | 8 | \begin{align*}6 + 8 = 14\end{align*} | |
Pentagonal prism | 10 | 15 | \begin{align*}10 + 7 = 17\end{align*} |
Do you see the pattern? When you add 2 to the number of edges, you get the sum of vertices and faces! The formula that summarizes this relationship is named after mathematician Leonhard Euler. Euler’s formula says, for any polyhedron:
Euler's Formula for Polyhedra:
\begin{align*}&\text{Vertices} + \text{Faces} = \text{Edges} + 2\\ & \qquad \quad \text{or}\\ &v + f = e + 2\end{align*}
Use Euler’s formula to find the number of edges, faces, or vertices in a polyhedron.
Regular Polyhedra
Polyhedra can be named and classified in a number of ways — by side, by angle, by base, by number of faces, and so on. Perhaps the most important classification is whether or not a polyhedron is regular or not. You will recall that a regular polygon is a polygon whose sides and angles are all congruent.
A polyhedron is regular if it has the following characteristics:
- All faces are the same.
- All faces are congruent regular polygons.
- The same number of faces meet at every vertex.
- The figure has no gaps or holes.
- The figure is convex — it has no indentations.
Reading Check:
1. True/False: A regular polyhedron has faces that are all different.
2. Fill in the blanks:
In a regular polyhedron, the same number of ____________________ meet at each _______________________.
3. True/False: A donut is not a regular polyhedron because it has a hole in the middle.
4. In your own words, describe a shape that is not convex:
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Example 4
Is a cube a regular polyhedron?
- All faces of a cube are regular polygons — squares.
- The cube is convex because it has no indented surfaces.
- The cube is simple because it has no gaps.
Therefore, a cube is a regular polyhedron.
A polyhedron is semi-regular if all of its faces are regular polygons and the same number of faces meet at every vertex. Semi-regular polyhedra often have two different kinds of faces, both of which are regular polygons.
A semi-regular polyhedron:
- has faces that are regular _________________________
- has the same number of ________________________ that meet at each vertex
- may have different kinds of _____________________
Prisms with a regular polygon base are one kind of semi-regular polyhedron.
Not all semi-regular polyhedra are prisms. An example of a non-prism is shown below:
Completely irregular (or not regular) polyhedra also exist. They are made of different kinds of regular and irregular polygons.
So now a question arises. Given that a polyhedron is regular if all of its faces are congruent regular polygons, it is convex and contains no gaps or holes. How many regular polyhedra actually exist?
In fact, you may be surprised to learn that only 5 regular polyhedra can be made. They are known as the Platonic (or noble) solids.
How many Platonic solids are there? __________
Note that no matter how you try, you can’t construct any other regular polyhedra besides the ones above.
Representing Solids: Nets
One way to represent a solid is to use a net. A net is a 2-dimensional picture that can be used to create a 3-dimensional solid. If you cut out a net you can fold it along the lines into a 3-D model of a figure. Here is an example of a net for a cube:
There is more than one way to make a net for a single figure. Here is another for a cube:
However, not all arrangements will create a cube:
Reading Check:
In your own words, what is a net?
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Example 5
What kind of figure does the net create? Draw the figure.
Remember, you can cut out the net and fold it along every line to make a 3-D figure.
The net above creates a box-shaped rectangular prism as shown below:
How many faces does the prism have? ________ (Hint: if you count the number of boxes in the net, it is the same as the number of faces in the 3-D figure!)
Example 6
What kind of net can you draw to represent the figure shown? Draw the net.
A net for the prism is shown. Other nets are also possible.
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