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11.1: Exploring Solids

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Identify different types of solids and their parts.
  • Use Euler’s formula and nets.

Review Queue

  1. Draw an octagon and identify the edges and vertices of the octagon. How many of each are there?
  2. Find the area of a square with 5 cm sides.
  3. Draw the following polygons.
    1. A convex pentagon.
    2. A concave nonagon.

Know What? Until now, we have only talked about two-dimensional, or flat, shapes. Copy the equilateral triangle to the right onto a piece of paper and cut it out. Fold on the dotted lines. What shape do these four equilateral triangles make?


Polyhedron: A 3-dimensional figure that is formed by polygons that enclose a region in space.

Each polygon in a polyhedron is a face.

The line segment where two faces intersect is an edge.

The point of intersection of two edges is a vertex.

Examples of polyhedrons include a cube, prism, or pyramid. Non-polyhedrons are cones, spheres, and cylinders because they have sides that are not polygons.

Prism: A polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles.

Pyramid: A polyhedron with one base and all the lateral sides meet at a common vertex.

All prisms and pyramids are named by their bases. So, the first prism would be a triangular prism and the first pyramid would be a hexagonal pyramid.

Example 1: Determine if the following solids are polyhedrons. If the solid is a polyhedron, name it and find the number of faces, edges and vertices each has.





a) The base is a triangle and all the sides are triangles, so this is a triangular pyramid. There are 4 faces, 6 edges and 4 vertices.

b) This solid is also a polyhedron. The bases are both pentagons, so it is a pentagonal prism. There are 7 faces, 15 edges, and 10 vertices.

c) The bases that are circles. Circles are not polygons, so it is not a polyhedron.

Euler’s Theorem

Let’s put our results from Example 1 into a table.

Faces Vertices Edges
Triangular Pyramid 4 4 6
Pentagonal Prism 7 10 15

Notice that faces + vertices is two more that the number of edges. This is called Euler’s Theorem, after the Swiss mathematician Leonhard Euler.

Euler’s Theorem: \begin{align*}F+V=E+2\end{align*}F+V=E+2.

\begin{align*}Faces + Vertices &= Edges +2\\ 5 + 6 &= 9 + 2\end{align*}Faces+Vertices5+6=Edges+2=9+2

Example 2: Find the number of faces, vertices, and edges in the octagonal prism.

Solution: There are 10 faces and 16 vertices. Use Euler’s Theorem, to solve for \begin{align*}E\end{align*}E.

\begin{align*}F + V &= E + 2\\ 10 + 16 &= E + 2\\ 24 &= E\end{align*}F+V10+1624=E+2=E+2=E

Example 3: In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?

Solution: Solve for \begin{align*}V\end{align*}V in Euler’s Theorem.

\begin{align*}F + V &= E + 2\\ 6 + V &= 10 + 2\\ V &= 6\end{align*}F+V6+VV=E+2=10+2=6

Example 4: A three-dimensional figure has 10 vertices, 5 faces, and 12 edges. Is it a polyhedron?

Solution: Plug in all three numbers into Euler’s Theorem.

\begin{align*}F + V &= E + 2\\ 5 + 10 &= 12 + 2\\ 15 & \neq 14\end{align*}F+V5+1015=E+2=12+214

Because the two sides are not equal, this figure is not a polyhedron.

Regular Polyhedra

Regular Polyhedron: A polyhedron where all the faces are congruent regular polygons.

All regular polyhedron are convex.

A concave polyhedron “caves in.”

There are only five regular polyhedra, called the Platonic solids.

Regular Tetrahedron: A 4-faced polyhedron and all the faces are equilateral triangles.

Cube: A 6-faced polyhedron and all the faces are squares.

Regular Octahedron: An 8-faced polyhedron and all the faces are equilateral triangles.

Regular Dodecahedron: A 12-faced polyhedron and all the faces are regular pentagons.

Regular Icosahedron: A 20-faced polyhedron and all the faces are equilateral triangles.


One way to “view” a three-dimensional figure in a two-dimensional plane, like in this text, is to use cross-sections.

Cross-Section: The intersection of a plane with a solid.

The cross-section of the peach plane and the tetrahedron is a triangle.

Example 5: What is the shape formed by the intersection of the plane and the regular octahedron?





a) Square

b) Rhombus

c) Hexagon


Net: An unfolded, flat representation of the sides of a three-dimensional shape.

Example 6: What kind of figure does this net create?

Solution: The net creates a rectangular prism.

Example 7: Draw a net of the right triangular prism below.

Solution: The net will have two triangles and three rectangles. The rectangles are different sizes and the two triangles are the same.

There are several different nets of any polyhedron. For example, this net could have the triangles anywhere along the top or bottom of the three rectangles. Click the site http://www.cs.mcgill.ca/~sqrt/unfold/unfolding.html to see a few animations of other nets.

Know What? Revisited The net of the shape is a regular tetrahedron.

Review Questions

  • Questions 1-8 are similar to Examples 2-4.
  • Questions 9-14 are similar to Example 1.
  • Questions 15-17 are similar to Example 5.
  • Questions 18-23 are similar to Example 7.
  • Questions 24-29 are similar to Example 6.
  • Question 30 uses Euler’s Theorem.

Complete the table using Euler’s Theorem.

Name Faces Edges Vertices
1. Rectangular Prism 6 12
2. Octagonal Pyramid 16 9
3. Regular Icosahedron 20 12
4. Cube 12 8
5. Triangular Pyramid 4 4
6. Octahedron 8 12
7. Heptagonal Prism 21 14
8. Triangular Prism 5 9

Determine if the following figures are polyhedra. If so, name the figure and find the number of faces, edges, and vertices.

Describe the cross section formed by the intersection of the plane and the solid.

Draw the net for the following solids.

Determine what shape is formed by the following nets.

  1. A truncated icosahedron is a polyhedron with 12 regular pentagonal faces and 20 regular hexagonal faces and 90 edges. This icosahedron closely resembles a soccer ball. How many vertices does it have? Explain your reasoning.

Review Queue Answers

  1. There are 8 vertices and 8 edges in an octagon.
  2. \begin{align*}5^2 = 25 \ cm^2\end{align*}52=25 cm2

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Feb 22, 2012
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