Lesson Plan: Advanced Polyhedra and Cylindrical Shapes

Lesson Summary

This 45-minute lesson introduces students to cylinders and advanced polyhedra, specifically the cylinder, dodecahedron, and icosahedron. Students will explore these shapes through animations, printable foldable nets, and hands-on activities, analyzing their faces, edges, and vertices.

Students will examine how curved surfaces in cylinders differ from flat-faced polyhedra and discuss the real-world applications of these shapes in engineering and design. The lesson also includes surface area calculations to reinforce mathematical problem-solving skills.

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

Lesson Objectives

• Identify and analyze nets of cylinders, dodecahedra, and icosahedra
• Predict how a net folds into a 3D figure
• Construct cylinders, dodecahedra, and icosahedra from printable nets
• Count and analyze the number of vertices, edges, and faces
• Calculate the surface area of cylinders and polyhedra using their nets

Common Core Standards

• CCSS.MATH.CONTENT.6.G.A.4 – Represent three-dimensional figures using nets made up of rectangles and polygons and use the nets to find the surface area.
• CCSS.MATH.CONTENT.7.G.B.6 – Solve real-world and mathematical problems involving area, volume, and surface area.

Prerequisite Skills

Before starting this lesson, students should be familiar with:

• Understanding of basic polyhedra and their properties.
• Ability to calculate the area of rectangles, triangles, and pentagons.
• Experience with surface area calculations for simpler 3D shapes.

Key Vocabulary

• Net – A two-dimensional pattern that can be folded to form a three-dimensional shape.
  • Multimedia Resource: https://www.media4math.com/library/43030/asset-preview
• Face – A flat surface on a three-dimensional figure.
  • Multimedia Resource: https://www.media4math.com/library/22039/asset-preview
• Edge – A line segment where two faces meet on a 3D shape.
  • Multimedia Resource: https://www.media4math.com/library/22031/asset-preview
• Vertex – A point where three or more edges meet.
  • Multimedia Resource: https://www.media4math.com/library/22207/asset-preview
• Cylinder – A 3D shape with two parallel circular bases and a curved surface connecting them.
• Dodecahedron – A polyhedron with 12 pentagonal faces, 30 edges, and 20 vertices.
• Icosahedron – A polyhedron with 20 triangular faces, 30 edges, and 12 vertices.

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 Exploring Cylindrical Packaging

• Show images of soda cans, batteries, and mailing tubes.
• Ask students: “Why are so many storage containers cylindrical instead of rectangular?”
• Discuss the advantages of cylindrical shapes in packaging and transportation.

Activity 2 Investigating Polyhedra in Nature

• Display images of geometric patterns in nature, such as crystals, geodesic domes, and beehives.
• Have students identify which natural structures resemble polyhedra.
• Discuss why nature often favors symmetrical polyhedral forms.

Activity 3 Digital Net Exploration

• Show an animated GIF from Media4Math.com demonstrating a net folding into a dodecahedron or icosahedron.
• Ask: “What do you notice about the way the faces come together?”
• Have students predict which nets will fold correctly before playing the animation.

Teach

Introduction

In this section, you will explore how 2D nets fold into 3D shapes. You will watch animations, construct models from printable foldable nets, and analyze the geometric properties of each shape. By the end of this section, you should understand how faces, edges, and vertices relate to nets and 3D figures.

Multimedia Resources

• A slide show for the net for a cylinder: 
• A slide show for the net for a dodecahedron: 
• A slide show for the net for an Icosahedron: 

Example 1 Cylinder

Let's start by exploring the cylinder! Follow these steps:

1. Watch the animation of a cylinder’s net folding into a 3D shape.
2. Cut out and fold the printable net to create a physical cylinder. Here's a link to a downloadable, printable net: Click here. 
3. As you fold, count the number of faces, edges, and vertices.

You should find:

• Faces: A cylinder has 3 faces (2 circular bases and 1 curved lateral surface).
• Edges: A cylinder has 2 edges (where the circular bases meet the curved surface).
• Vertices: A cylinder has 0 vertices (since it has no sharp corners).

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

Example 2 Dodecahedron

Now, let's examine a dodecahedron!

1. Watch the animation of a dodecahedron’s net folding.
2. Cut out the printable net and fold it into a dodecahedron. Here's a link to a downloadable, printable net: Click here. 
3. Count the number of faces, edges, and vertices as you fold.

You should find:

• Faces: 12 pentagonal faces.
• Edges: 30 edges.
• Vertices: 20 vertices.

Think About It: Why do polyhedra like the dodecahedron appear in molecular structures?

Example 3 Icosahedron

Finally, let's investigate the icosahedron!

1. Watch the animation of an icosahedron’s net folding.
2. Cut out the printable net and fold it into an icosahedron. Here's a link to a downloadable, printable net: Click here. 
3. Count the number of faces, edges, and vertices.

You should find:

• Faces: 20 triangular faces.
• Edges: 30 edges.
• Vertices: 12 vertices.

Think About It: How does an icosahedron compare to a dodecahedron in terms of shape and structure?

Example 4 Surface Area of a Cylinder

Now that we’ve built a cylinder, let's calculate its surface area!

A cylinder consists of two circular bases and a curved lateral surface. The total surface area formula is:

\[ SA = 2\pi r^2 + 2\pi rh \]

Where:

• \( r \) is the radius of the circular base.
• \( h \) is the height of the cylinder.

Let's Try It!

Suppose a cylinder has a radius of 4 cm and a height of 10 cm. Find its surface area.

\[ SA = 2\pi (4)^2 + 2\pi (4)(10) \] \[ SA = 2\pi (16) + 2\pi (40) \] \[ SA = 32\pi + 80\pi \] \[ SA = 112\pi \text{ cm}^2 \]

Final Answer: The surface area of this cylinder is 112π cm² or approximately 351.86 cm².

Think About It: How does increasing the height affect the surface area?

Example 5 Surface Area of a Dodecahedron

Now, let's calculate the surface area of a dodecahedron.

A regular dodecahedron consists of 12 pentagonal faces. The surface area formula is:

\[ SA = 12 \times \text{area of one pentagon} \]

To find the area of one pentagonal face, use the formula:

\[ A = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} e^2 \]

Where:

• \( e \) is the length of one edge.

Let's Try It!

Suppose a dodecahedron has an edge length of 5 cm. Find its surface area.

\[ A = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} (5)^2 \] \[ A \approx 43.01 \text{ cm}^2 \] \[ SA = 12 \times 43.01 \] \[ SA \approx 516.12 \text{ cm}^2 \]

Final Answer: The surface area of this dodecahedron is approximately 516.12 cm².

Think About It: How does the number of faces affect the surface area of a polyhedron?

Review

Lesson Recap

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

• Nets are 2D representations of 3D shapes that can be folded to form solid figures.
• Each 3D shape has a specific number of faces, edges, and vertices.
• Surface area is found by calculating the total area of all faces of a 3D shape.
• Cylinders have curved surfaces that distinguish them from polyhedra.
• Dodecahedra and icosahedra are complex polyhedra found in both mathematics and nature.
• Nets and polyhedral structures are used in architecture, packaging, and engineering.

Example 1 Cylinders in Packaging

Think about why soda cans, batteries, and other containers are often cylindrical.

Let’s explore!

1. What advantages do cylinders have in terms of strength and storage efficiency?
2. How do curved surfaces reduce material waste?
3. Compare the storage capacity of a cylindrical can versus a rectangular box.

Think About It: How would shipping costs change if companies used cubes instead of cylinders for cans?

Example 2 Surface Area of an Icosahedron

An icosahedron consists of 20 triangular faces. To find its surface area, use the formula:

\[ SA = 20 \times \text{area of one triangle} \]

Let's Try It!

Suppose each triangular face has a base of 6 cm and a height of 5 cm. Find the surface area.

\[ SA = 20 \times \left( \frac{1}{2} (6 \times 5) \right) \] \[ SA = 20 \times (15) \] \[ SA = 300 \text{ cm}^2 \]

Final Answer: The surface area of the icosahedron is 300 cm².

Think About It: Why are icosahedral structures common in viral and molecular design?

Example 3 Polyhedral Structures in Nature

Many natural formations resemble polyhedral structures.

• Viruses, such as the common cold virus, often have icosahedral shapes for stability.
• Crystals naturally form dodecahedral and other polyhedral structures.
• Geodesic domes, used in architecture, mimic polyhedral frameworks.

Think About It: How do engineers use polyhedral principles to design earthquake-resistant buildings?

Quiz

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

1. What is a net in geometry?
2. How many faces does a cylinder have?
3. How many faces, edges, and vertices does a dodecahedron have?
4. How many triangular faces does an icosahedron have?
5. What is the formula for the surface area of a cylinder?
6. A cylinder has a radius of 5 cm and a height of 12 cm. Calculate its surface area.
7. A dodecahedron has 12 pentagonal faces. If each pentagon has an area of 10 cm², what is its total surface area?
8. Why do engineers use icosahedral structures in dome designs?
9. How does a cylinder differ from a prism?
10. What real-world objects resemble an icosahedron?

Answer Key

1. A net is a 2D pattern that folds into a 3D shape.
2. 3 faces (2 circular bases, 1 curved lateral surface).
3. 12 faces, 30 edges, 20 vertices.
4. 20 triangular faces.
5. \( SA = 2\pi r^2 + 2\pi rh \).
6. \( SA = 2\pi (5)^2 + 2\pi (5)(12) = 50\pi + 120\pi = 170\pi \) cm².
7. \( SA = 12 \times 10 = 120 \) cm².
8. Icosahedral structures distribute weight evenly, making them stable for dome construction.
9. A prism has only flat faces, while a cylinder has a curved lateral surface.
10. Examples: Some dice, molecules, and viral capsids.