Lesson Plan: Extensions and Real-World Applications of Nets and 3D Shapes

Lesson Summary

This 45-minute lesson extends students’ understanding of nets and 3D figures by exploring advanced applications, geometric transformations, and STEM connections. Students will analyze Euler’s Formula, explore geometric transformations in nets, and apply optimization strategies in packaging design to minimize material waste.

Through hands-on activities, students will discover how engineers, architects, and designers use nets to model 3D structures, and they will explore digital tools such as GeoGebra and Tinkercad to manipulate 3D shapes. This lesson reinforces real-world problem-solving skills and provides insight into how mathematics is applied in technology and engineering.

Multimedia resources from Media4Math.com will support this lesson. A 10-question quiz with an answer key is included for assessment.

Lesson Objectives

• Apply Euler’s Formula to verify the relationship between vertices, edges, and faces of different polyhedra.
• Analyze geometric transformations in nets, including rotations and reflections.
• Investigate optimization strategies for designing efficient packaging.
• Explore STEM applications by examining how engineers use nets in 3D modeling and design.
• Use digital tools to manipulate and visualize nets and 3D shapes interactively.

Common Core Standards

• CCSS.MATH.CONTENT.7.G.A.2 – Draw and describe geometric figures, including transformations such as reflections and rotations.
• CCSS.MATH.CONTENT.8.G.A.4 – Understand that rotations, reflections, and translations preserve distance and angle measures.
• CCSS.MATH.CONTENT.HSG.MG.A.3 – Apply geometric methods to solve design problems related to packaging and modeling.

Prerequisite Skills

Before starting this lesson, students should be familiar with:

• Basic properties of polyhedra (faces, edges, vertices).
• Surface area calculations for prisms, pyramids, and polyhedra.
• Understanding of geometric transformations, including reflections and rotations.
• Basic knowledge of 3D modeling software (optional).

Key Vocabulary

• Euler’s Formula – A formula that relates the number of vertices (V), edges (E), and faces (F) in a polyhedron:

\[ V - E + F = 2 \]

• Geometric Transformation – A movement of a shape, including reflections, rotations, and translations, that preserves size and shape.
• Optimization – The process of minimizing material waste while achieving the best design efficiency.
• 3D Modeling – The use of digital tools to create and manipulate three-dimensional structures.
• Tinkercad – An online 3D design and modeling tool used for creating digital models.
• GeoGebra – A dynamic geometry software that allows students to interactively explore nets and transformations.

Multimedia Resources from Media4Math.com

• A collection of definition cards on the topic of 3D Geometry: https://www.media4math.com/Definitions--3DGeometry
• A collection of video definitions on the topic of 3D Geometry: https://www.media4math.com/MathVideoCollection--3DGeoVocabulary

Warm Up Activities

Choose from one or more of these activities.

Activity 1 Reviewing 3D Nets

• Provide students with a set of printed or digital 3D nets (e.g., cube, rectangular prism, tetrahedron).
• Ask students to predict which 3D shape each net will fold into.
• Show animations of nets folding into their respective 3D figures.
• Discuss the characteristics of each net, such as how many faces, edges, and vertices they will form.
• You can use these slide show to show the animations and nets above:
  • A slide show for the net for a cube: https://www.media4math.com/library/slideshow/net-cube
  • A slide show for the net for a rectangular prism: https://www.media4math.com/library/slideshow/net-rectangular-prism

Activity 2 Identifying 3D Shapes from Top, Front, and Side Views

Top	Side	Front

• Show students three different views (top, front, and side) of a hidden 3D shape.
• Ask: "Based on these views, what 3D shape do you think this represents?"
• The table above shows a cube, rectangular prism, and triangular prism.
• Discuss how engineers and designers use orthographic projections to visualize 3D objects.

Activity 3 Exploring the Motions Involved in Constructing a 3D Shape

• Ask: “What motions do you use when folding a net into a 3D shape?”
• Have students physically fold a paper net of a cube or pyramid.
• Guide them to describe the types of movements involved:
  • Rotations: Folding a face over an edge.
  • Reflections: How faces line up symmetrically.
  • Translations: Moving a face to align it before attaching.
• Discuss how these motions are important in engineering and 3D printing.

Teach

Introduction

In this section, we will explore how Euler’s Formula helps us understand the structure of polyhedra. This formula states that for any convex polyhedron:

\[ V - E + F = 2 \]

where:

• V = number of vertices
• E = number of edges
• F = number of faces

We will apply Euler’s Formula to different shapes and explore real-world applications of nets, transformations, and optimizations in design.

Example 1 Applying Euler’s Formula to a Cube

Problem: Verify Euler’s Formula for a cube.

1. A cube has:
   • Vertices: 8
   • Edges: 12
   • Faces: 6
2. Substituting into Euler’s Formula:

\[ 8 - 12 + 6 = 2 \]

Answer: Since Euler’s Formula holds, the cube follows this geometric rule.

Think About It: Why does Euler’s Formula work for all convex polyhedra?

Example 2 Applying Euler’s Formula to a Tetrahedron

Problem: Verify Euler’s Formula for a tetrahedron.

1. A tetrahedron has:
   • Vertices: 4
   • Edges: 6
   • Faces: 4
2. Substituting into Euler’s Formula:

\[ 4 - 6 + 4 = 2 \]

Answer: Euler’s Formula is valid for a tetrahedron.

Think About It: What do you notice about the number of faces, edges, and vertices compared to the cube?

Example 3 Applying Euler’s Formula to an Octahedron

Problem: Verify Euler’s Formula for an octahedron.

1. An octahedron has:
   • Vertices: 6
   • Edges: 12
   • Faces: 8
2. Substituting into Euler’s Formula:

\[ 6 - 12 + 8 = 2 \]

Answer: Euler’s Formula works for an octahedron.

Think About It: The number of edges in an octahedron is the same as in a cube. Why?

Example 4 Applying Euler’s Formula to a Rectangular Prism

Problem: Verify Euler’s Formula for a rectangular prism.

1. A rectangular prism has:
   • Vertices: 8
   • Edges: 12
   • Faces: 6
2. Substituting into Euler’s Formula:

\[ 8 - 12 + 6 = 2 \]

Answer: Euler’s Formula is valid for a rectangular prism.

Think About It: How does a rectangular prism compare to a cube in terms of faces, edges, and vertices?

Example 5 Optimization in Packaging Design

Problem: A company is designing a cereal box with a total surface area of 600 cm². The box is a rectangular prism with a square base and a height of h. What dimensions will maximize the box’s volume?

Step 1: Define the Variables

• Let x be the length of one side of the square base.
• Let h be the height of the box.
• The total surface area of the box is 600 cm².

Step 2: Write the Surface Area Equation

The total surface area of a rectangular prism with a square base is given by:

\[ SA = 2x^2 + 4xh \]

Since the total available material is 600 cm²:

\[ 600 = 2x^2 + 4xh \]

Step 3: Express h in Terms of x

Rearrange the equation to solve for h:

\[ h = \frac{600 - 2x^2}{4x} \]

Step 4: Write the Volume Equation

The volume of the box is given by:

\[ V = x^2h \]

Substituting our expression for h:

\[ V(x) = x^2 \times \frac{600 - 2x^2}{4x} \] \[ V(x) = \frac{600x - 2x^3}{4} \]

Step 5: Find the Maximum Volume by Using a Table

To find the optimal value of x, we test different values and compute the volume.

The largest volume occurs when x = 10 cm. Solving for h:

\[ h = \frac{600 - 2(10)^2}{4(10)} \] \[ h = \frac{600 - 200}{40} = \frac{400}{40} = 10 \text{ cm} \]

This is also confirmed by the graph.

Final Answer:

• Base: 10 cm × 10 cm
• Height: 10 cm
• Maximum Volume: 1000 cm³

Think About It: Why does the volume decrease when x gets too large?

Review

Lesson Recap

Let’s summarize what we’ve learned in this lesson:

• Euler’s Formula establishes a relationship between the vertices, edges, and faces of polyhedra.
• Geometric transformations (rotations, reflections) do not change the final 3D structure.
• Optimizing net designs reduces material waste in packaging and manufacturing.
• Nets are widely used in architecture, engineering, and space exploration.

Example 1 Why Are Most Shipping Containers Rectangular Prisms?

Problem: Shipping companies mostly use rectangular prisms for packaging. Why?

Step 1: Consider Stackability

• Rectangular prisms fit neatly together with no wasted space between them.
• Cylindrical or irregularly shaped packages leave gaps, which reduces efficiency.

Step 2: Compare Volume and Surface Area

• A rectangular box maximizes the usable space while keeping the surface area minimal.
• Less surface area means less packaging material is needed, reducing costs.

Step 3: Consider Transportation Efficiency

• Trucks and cargo ships use boxes that can stack perfectly to avoid wasted space.

Final Answer: Rectangular boxes are used because they stack efficiently, reduce wasted space, and minimize material costs.

Example 2 How Does Net Orientation Affect Paper Model Design?

Problem: Two students are building a cube from a net. One student uses a cross-shaped net, while the other uses a row-shaped net. Will both nets fold into a cube? Which one is more efficient?

Step 1: Identify the Shapes in Each Net

• Net A (Cross-Shaped Net): Six squares arranged in a cross pattern.
• Net B (Staggered Net): A variation on the cross pattern.

Step 2: Predict How Each Net Folds

• Net A: The central square forms the base, while the four attached squares fold up to form the sides. The final square folds over as the top.
• Net B: The same square formes the base and there are two flaps that require different types of folds. 

Step 3: Compare Efficiency

• The cross-shaped net folds into a cube simply, with minimal adjustments.
• The staggered net also folds into a cube with a little more effort.

Final Answer:

The cross-shaped net is more efficient because it naturally forms a cube with fewer folds.

Think About It: Why do packaging companies prefer nets that fold efficiently?

Example 3 Surface Area Challenge: How Does Optimizing Net Shape Reduce Material Waste?

Problem: A company wants to redesign a juice box to reduce material use while keeping the volume the same. How can they do this?

Step 1: Identify Key Measurements

• The current juice box has a base of 5 cm × 5 cm and a height of 12 cm.
• The surface area formula for a rectangular prism is:

\[ SA = 2lw + 2lh + 2wh \]

Step 2: Calculate the Current Surface Area

\[ SA = 2(5)(5) + 2(5)(12) + 2(5)(12) \] \[ SA = 50 + 120 + 120 = 290 \text{ cm}^2 \]

Step 3: Find a More Efficient Shape

• If the company changes the base to 6 cm × 6 cm and adjusts the height to keep the volume the same, how does the surface area change?

\[ SA = 2(6)(6) + 2(6)(\frac{300}{36}) + 2(6)(\frac{300}{36}) \] \[ SA = 2(36) + 2(6)(8.33) + 2(6)(8.33) \] \[ SA = 72 + 99.96 + 99.96 = 271.92 \text{ cm}^2 \]

Step 4: Compare Results

• The new design reduces the surface area from 290 cm² to 271.92 cm², saving material while keeping the same volume.

Final Answer: By adjusting the dimensions, the company saves 18.08 cm² of material per box.

Quiz

Directions: Answer the following questions. Show your work where necessary.

1. What is Euler’s Formula for polyhedra?
2. How many edges does a cube have?
3. Why does Euler’s Formula not work for cylinders?
4. How do rotations and reflections affect the way a net folds into a 3D shape?
5. What is an example of optimization in packaging?
6. A dodecahedron has 12 pentagonal faces. If each pentagon has an area of 15 cm², what is the total surface area?
7. A net of a cube is rotated 90 degrees. Will it still fold into the same shape? Why or why not?
8. How do architects use nets in building designs?
9. A box manufacturer wants to design a container that uses the least material while maximizing volume. What factors should they consider?

Answer Key

1. \( V - E + F = 2 \).
2. 12 edges.
3. Cylinders do not have polygonal faces, making the formula invalid.
4. They change how the shape folds but keep the structure the same.
5. Reducing material waste in cereal box designs.
6. \( SA = 12 \times 15 = 180 \) cm².
7. Yes, because rotations do not change the net’s shape.
8. Nets help architects plan and visualize how structures will be built.
9. They must balance material use, shape efficiency, and strength.