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Algebra 1

1.2.2 Modeling with Equations to Find Edges in Platonic Solids

Algebra 11.2.2 Modeling with Equations to Find Edges in Platonic Solids

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Activity

These three figures are called Platonic solids.

Tetrahedron Cube Dodecahedron
A simple black outline drawing of a three-dimensional octahedron, showing all its triangular faces with one dashed line indicating a hidden edge inside the shape. A black outline drawing of a transparent cube, showing solid front and top edges and dashed lines for hidden edges in the back. Line drawing of a dodecahedron, a three-dimensional geometric shape with twelve pentagonal faces, shown with solid and dashed lines to indicate visible and hidden edges.
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.

1.

Number of faces.

2.

Number of edges.

3.

Number of vertices.

There are some interesting relationships between the number of faces (FF), edges (EE), and vertices (VV) in all Platonic solids.

4.

What do you notice about the relationships between the numbers of faces, edges, and vertices of the Platonic solids?

5.

What do you wonder about the relationships between the numbers of faces, edges, and vertices of the Platonic solids?

6.

There is a relationship that can be expressed with an equation using all three values: F,V,EF,V,E. Write the equation that represents how the three values are related.

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?

b. The icosahedron has how many 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?

b. The icosahedron has how many vertices?

3. For all 5 Platonic solids, determine how many faces meet 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.

  1. 90 a b = 180
  2. a + b + 90 = 180
  3. a + b 90 = 180
  4. a b + 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 xx = the first integer. There are two more numbers that follow xx.
Each number is 1 more than the number before it: x+1x+1 and x+2x+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 xx, x+1x+1, and x+2x+2 is 372.

Step 4 - Translate the sentence into an equation.
x+(x+1)+(x+2)=372x+(x+1)+(x+2)=372
3x+3=3723x+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 xx = the first integer. There are two more ODD numbers that follow xx.
Each number is 2 more than the number before it: x+2x+2 and x+4x+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 xx, x+2x+2, and x+4x+4 is 93.

Step 4 - Translate the sentence into an equation.
x+(x+2)+(x+4)=93x+(x+2)+(x+4)=93
3x+6=933x+6=93

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.

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