Activity
These three figures are called Platonic solids.
Tetrahedron | Cube | Dodecahedron |
|
|
|
faces | vertices | edges | |
tetrahedron | 4 | 4 | 6 |
cube | |||
dodecahedron | 12 | 20 | 30 |
The table shows the number of faces, vertices, and edges for the tetrahedron and dodecahedron.
Determine the following values regarding the cube.
Number of faces.
6.
Number of edges.
8.
Number of vertices.
12.
There are some interesting relationships between the number of faces (), edges (), and vertices () in all Platonic solids.
What do you notice about the relationships between the numbers of faces, edges, and vertices of the Platonic solids?
Your answer may vary, but here are some examples.
- I noticed all the numbers are even.
- I noticed the number of faces is less than or equal to the number of vertices, or
- I noticed the number of edges is always greater than the number of faces, or .
What do you wonder about the relationships between the numbers of faces, edges, and vertices of the Platonic solids?
Your answer may vary, but here are some examples.
- I wonder if the number of edges is always less than the sum of the number of faces and the number of vertices, or .
- I wonder if there is an equation that connects all of the variables.
There is a relationship that can be expressed with an equation using all three values: . Write the equation that represents how the three values are related.
Compare your answer:
The number of edges is 2 less than the sum of the other two numbers, or .
Are you ready for more?
Extending Your Thinking
There are two more Platonic solids: an octahedron, which has 8 faces that are all triangles, and an icosahedron, which has 20 faces that are all triangles.
1. How many edges would each of these solids have? (Keep in mind that each edge is used in two faces.)
a. The octahedron has how many edges?
The octahedron has 12 edges.
b. The icosahedron has how many edges?
The icosahedron has 30 edges.
2. Use your discoveries from the activity to determine how many vertices each of these solids would have.
a. The octahedron has how many vertices?
The octahedron has 6 vertices.
b. The icosahedron has how many vertices?
The icosahedron has 30 vertices.
3. For all 5 Platonic solids, determine how many faces meet at each vertex.
The tetrahedron, cube, and dodecahedron have 3 faces meeting at each vertex. The octahedron has 4 faces meeting at each vertex. The icosahedron has 5 faces meeting at each vertex.
Self Check
Additonal Resources
Writing Equations Using Symbols
How to write statements using algebraic symbols
Step 1 - Read the problem.
Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.
Step 3 - Write a sentence using the relationship among the values.
Step 4 - Translate the sentence into an equation.
Remember that variables represent values in the problem that can change.
Let's examine two examples of how to write statements algebraically using symbols.
Example 1
We want to express the following statement using symbolic language:
The sum of three consecutive integers is 372.
Step 1 - Read the problem.
The sum of three consecutive integers is 372.
Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.
Let = the first integer. There are two more numbers that follow .
Each number is 1 more than the number before it: and
The sum of all three numbers is 372.
Step 3 - Write a sentence using the relationship among the values.
The sum of , , and is 372.
Step 4 - Translate the sentence into an equation.
Example 2
We want to express the following statement using symbolic language:
The sum of three consecutive odd integers is 93.
Step 1 - Read the problem.
The sum of three consecutive odd integers is 93.
Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.
Let = the first integer. There are two more ODD numbers that follow .
Each number is 2 more than the number before it: and
The sum of all three numbers is 93.
Step 3 - Write a sentence using the relationship among the values.
The sum of , , and is 93.
Step 4 - Translate the sentence into an equation.
Try it
Try It: Writing Equations Using Symbols
Now it’s your turn. Write the statement below algebraically using symbols.
The sum of three consecutive integers is 1,623.
Here is how to express this statement using symbols.