Although polyhedra are generally named by the number of faces that make up the figure, it is common to see qualifier of the face shape within the name.

Likewise, in naming three-dimensional figures that are not regular polyhedron, it is critical to know the shape of the base. The name of the shape of the base will be used in naming the solid and the net of the solid.

Source: Leonhard Euler, Jakob Emmanuel Handmann, Wikimedia Commons

Euler’s Theorem tells us the relationship between the number of faces, edges, and vertices in any polyhedron. Remember that a polyhedron is a three-dimensional figure whose faces are all polygons, and polygons are planar shapes with closed edges and no curved edges.

Will Euler’s Theorem still apply to a polyhedron that is not regular?

Look at the table below to see information about each polyhedron.

Conclusion Questions

• Why do you think Euler’s Theorem still applies to polyhedra that are not regular?

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  How are regular polyhedra different from the 3 polyhedra that you investigated in the applet?

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  Although these nets do not fold into regular polyhedra, they are still polyhedra. The relationship that is expressed in Euler’s Theorem is regarding the relationship of polygons that create the polyhedron. Even though the orientation of the shape may change, as seen by the nets, the attributes and characteristics of the shape do not.

• To what nets of three-dimensional figures, if any, will Euler’s Theorem not apply?

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  What three-dimensional figures can you think of that are not composed entirely of faces that are polygons?

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  Euler’s Theorem will not apply to nets of cylinders and cones because the shapes in the nets themselves are not polyhedra, meaning they are not formed by polygons.

Pause and Reflect

How does using a net help you to visually recognize faces, edges, and vertices of a three-dimensional figure?

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A net lays out on a flat plane each face of the three-dimensional figure. The edges of each face represent an edge of the three-dimensional figure. Vertices are represented where edges meet. However, the number of edges and vertices in a net may not match the number of edges and vertices in the actual three-dimensional figure because some edges and vertices are separated into component parts.

Practice

1. What 3-D figure will the net below make when it is folded?

   

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   How many faces are present in the net? What shapes are the faces?

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   Cube

2. For the figure whose net is shown below, identify the figure, the number of faces, the number of edges, and the number of vertices.

   

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   How many faces are present in the net? What shapes are the faces?

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   Name of Solid: square pyramid
   Number of Faces: 5
   Number of Edges: 8
   Number of Vertices: 5

3. Sheila constructed the net shown below.

   

   How many faces, edges, and vertices does the three-dimensional figure represented by the net have?

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   What type of figure is represented by the net?

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   Number of Faces: 8
   Number of Edges: 18
   Number of Vertices: 12