Modeling with Equations to Find Edges in Platonic Solids

1.2.2 Modeling with Equations to Find Edges in Platonic Solids

1.2.2 • Modeling with Equations to Find Edges in Platonic Solids

Activity

These three figures are called Platonic solids.

Tetrahedron	Cube	Dodecahedron

Illustrations of a Tetrahedron, Cube, and Dodecahedron, showcasing various geometric shapes.

	faces	vertices	edges
tetrahedron	4	4	6
cube
dodecahedron	12	20	30

Counts of faces, vertices, and edges for various polyhedra.

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.

Number of edges.

Number of vertices.

12.

There are some interesting relationships between the number of faces ( $F$ ), edges ( $E$ ), and vertices ( $V$ ) 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 $F \leq V$
I noticed the number of edges is always greater than the number of faces, or $E > F$ .

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 $E < F + V$ .
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: $F, V, E$ . 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 $V + F - 2 = E$ .

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

The sum of the measures of the angles of a triangle is $180°$ . The right triangle shown has angle measures $a$ , $b$ , and $90°$ . Which equation could model this relationship?

A right triangle with angle measures a, b, and ninety degrees.

$90ab=180$
$a + b + 90 = 180$
$a + b - 90 =180$
$ab + 90 = 180$

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 $x$ = the first integer. There are two more numbers that follow $x$ .
Each number is 1 more than the number before it: $x + 1$ and $x + 2$
The sum of all three numbers is 372.

A number line showing points labeled x, x+1, and x+2. Two curved arrows point from x to x+1 (labeled +1) and from x to x+2 (labeled +2).

Step 3 - Write a sentence using the relationship among the values.
The sum of $x$ , $x + 1$ , and $x + 2$ is 372.

Step 4 - Translate the sentence into an equation.
$x + (x + 1) + (x + 2) = 372$
$3 x + 3 = 372$

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 $x$ = the first integer. There are two more ODD numbers that follow $x$ .
Each number is 2 more than the number before it: $x + 2$ and $x + 4$
The sum of all three numbers is 93.

A number line showing points labeled x, x+2, and x+4. Arrows connect x to x+2 with +2 and x to x+4 with +4. The labels even # appear under x and x+2.

Step 3 - Write a sentence using the relationship among the values.
The sum of $x$ , $x + 2$ , and $x + 4$ is 93.

Step 4 - Translate the sentence into an equation.
$x + (x + 2) + (x + 4) = 93$
$3 x + 6 = 93$

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.

$x + (x + 1) + (x + 2) = 1, 623$

$3 x + 3 = 1, 623$