What if you were given a solid three-dimensional figure, like a carton of ice cream? How could you determine how the faces, vertices, and edges of that figure are related? After completing this Concept, you'll be able to use Euler's Theorem to answer that question.
Watch This
CK-12 Foundation: Chapter11PolyhedronsA
Learn more about 3-D Solid Properties by watching the video at this link.
Guidance
A polyhedron is a 3-dimensional figure that is formed by polygons that enclose a region in space. Each polygon in a polyhedron is called a face. The line segment where two faces intersect is called an edge and the point of intersection of two edges is a vertex. There are no gaps between the edges or vertices in a polyhedron. Examples of polyhedrons include a cube, prism, or pyramid. Non-polyhedrons are cones, spheres, and cylinders because they have sides that are not polygons.
A prism is a polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles. Prisms are explored in further detail in another Concept.
A pyramid is a polyhedron with one base and all the lateral sides meet at a common vertex. The lateral sides are triangles. Pyramids are explored in further detail in another Concept.
All prisms and pyramids are named by their bases. So, the first prism would be a triangular prism and the second would be an octagonal prism. The first pyramid would be a hexagonal pyramid and the second would be a square pyramid. The lateral faces of a pyramid are always triangles.
Euler’s Theorem states that the number of faces \begin{align*}(F)\end{align*}, vertices \begin{align*}(V)\end{align*}, and edges \begin{align*}(E)\end{align*} of a polyhedron can be related such that \begin{align*}F+V=E+2\end{align*}.
A regular polyhedron is a polyhedron where all the faces are congruent regular polygons. There are five regular polyhedra called the Platonic solids, after the Greek philosopher Plato. These five solids are significant because they are the only five regular polyhedra. There are only five because the sum of the measures of the angles that meet at each vertex must be less than \begin{align*}360^\circ\end{align*}. Therefore the only combinations are 3, 4 or 5 triangles at each vertex, 3 squares at each vertex or 3 pentagons. Each of these polyhedra have a name based on the number of sides, except the cube.
- Regular Tetrahedron: A 4-faced polyhedron where all the faces are equilateral triangles.
- Cube: A 6-faced polyhedron where all the faces are squares.
- Regular Octahedron: An 8-faced polyhedron where all the faces are equilateral triangles.
- Regular Dodecahedron: A 12-faced polyhedron where all the faces are regular pentagons.
- Regular Icosahedron: A 20-faced polyhedron where all the faces are equilateral triangles.
Example A
Determine if the following solids are polyhedrons. If the solid is a polyhedron, name it and determine the number of faces, edges and vertices each has.
a)
b)
c)
Solutions:
a) The base is a triangle and all the sides are triangles, so this is a polyhedron, a triangular pyramid. There are 4 faces, 6 edges and 4 vertices.
b) This solid is also a polyhedron because all the faces are polygons. The bases are both pentagons, so it is a pentagonal prism. There are 7 faces, 15 edges, and 10 vertices.
c) This is a cylinder and has bases that are circles. Circles are not polygons, so it is not a polyhedron.
Example B
Find the number of faces, vertices, and edges in the octagonal prism.
Because this is a polyhedron, we can use Euler’s Theorem to find either the number of faces, vertices or edges. It is easiest to count the faces, there are 10 faces. If we count the vertices, there are 16. Using this, we can solve for \begin{align*}E\end{align*} in Euler’s Theorem.
\begin{align*}F + V & = E + 2\\ 10 + 16 & = E + 2\\ 24 & = E \qquad \text{There are} \ 24 \ \text{edges}.\end{align*}
Example C
In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?
Solve for \begin{align*}V\end{align*} in Euler’s Theorem.
\begin{align*}F + V & = E + 2\\ 6 + V & = 10 + 2\\ V & = 6 \qquad \text{There are} \ 6 \ \text{vertices}.\end{align*}
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter11PolyhedronsB
Vocabulary
A polyhedron is 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. A regular polyhedron is a polyhedron where all the faces are congruent regular polygons.
Guided Practice
1. In a six-faced polyhedron, there are 10 edges. How many vertices does the polyhedron have?
2. Markus counts the edges, faces, and vertices of a polyhedron. He comes up with 10 vertices, 5 faces, and 12 edges. Did he make a mistake?
3. Is this a polyhedron? Explain.
Answers:
1. Solve for \begin{align*}V\end{align*} in Euler’s Theorem.
\begin{align*}F + V &= E + 2\\ 6 + V &= 10 + 2\\ V &= 6\end{align*}
Therefore, there are 6 vertices.
2. Plug all three numbers into Euler’s Theorem.
\begin{align*}F + V &= E + 2\\ 5 + 10 &= 12 + 2\\ 15 & \neq 14\end{align*}
Because the two sides are not equal, Markus made a mistake.
3. All of the faces are polygons, so this is a polyhedron. Notice that even though not all of the faces are regular polygons, the number of faces, vertices, and edges still works with Euler's Theorem.
Practice
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.
- 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.
For problems 15-17, we are going to connect the Platonic Solids to probability. A six sided die is the shape of a cube. The probability of any one side landing face up is \begin{align*}\frac{1}{6}\end{align*} because each of the six faces is congruent to each other.
- What shape would we make a die with 12 faces? If we number these faces 1 to 12, and each face has the same likelihood of landing face up, what is the probability of rolling a multiple of three?
- I have a die that is a regular octahedron. Each face is labeled with a consecutive prime number starting with 2. What is the largest prime number on my die?
- Challenge Rebecca wants to design a new die. She wants it to have one red face. The other faces will be yellow, blue or green. How many faces should her die have and how many of each color does it need so that: the probability of rolling yellow is eight times the probability of rolling red, the probability of rolling green is half the probability of rolling yellow and the probability of rolling blue is seven times the probability of rolling red?