Lesson Plan: Pyramids, Tetrahedra, and Octahedra

Lesson Summary

This 45-minute lesson introduces students to the geometry of pyramids and tetrahedra using 2D nets and 3D models. Students will watch animations, construct foldable nets, and analyze the faces, edges, and vertices of tetrahedra, square pyramids, and octahedra.

Through hands-on exploration, students will compare pyramids to prisms, discussing why triangular structures are so stable and how pyramids are used in architecture and engineering. Students will also calculate surface areas using nets, applying mathematical formulas to real-world problems.

Multimedia resources from Media4Math.com are integrated throughout the lesson. A 10-question quiz with an answer key is included for assessment.

Lesson Objectives

• Identify and analyze nets of pyramids and tetrahedra
• Predict how a net folds into a 3D pyramid or tetrahedron
• Construct tetrahedra, square pyramids, and octahedra from printable nets
• Count and analyze the number of vertices, edges, and faces
• Calculate the surface area of pyramids and tetrahedra using their nets

Common Core Standards

• CCSS.MATH.CONTENT.6.G.A.4 – Represent three-dimensional figures using nets made up of triangles 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 2D triangles and their properties
• Ability to use basic area formulas for triangles
• Familiarity with geometric terms (faces, edges, vertices)
• Experience calculating surface area of rectangular prisms

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
• Surface Area – The total area of all faces of a three-dimensional figure.
• Pyramid – A 3D shape with a polygonal base and triangular faces that meet at a single point.
• Tetrahedron – A 3D shape with four triangular faces, also known as a triangular pyramid.
• Octahedron – A 3D shape made of eight triangular faces, resembling two pyramids joined at the base.

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 Pyramid Mystery Challenge

• Show images of famous pyramids (e.g., Egyptian pyramids, Mayan pyramids).
• Ask: “What do these structures have in common? Why are pyramids so stable?”
• Have students sketch a possible net for a square pyramid.
• Discuss how different nets can form the same pyramid.

Activity 2 Exploring 3D Stability

• Provide toothpicks and marshmallows to construct a cube and a tetrahedron.
• Ask students to compare which shape is stronger and more stable.
• Discuss why tetrahedra are common in strong structures (e.g., bridges, trusses).

Activity 3 Digital Net Exploration

• Show an animated GIF from Media4Math.com that demonstrates a pyramid net folding into a 3D shape.
• Ask: “What would happen if one triangular face were missing?”
• 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 tetrahedron: https://www.media4math.com/library/slideshow/net-tetrahedron
• A slide show for the net for a square pyramid: https://www.media4math.com/library/slideshow/net-square-pyramid
• A slide show for the net for an octahedron: https://www.media4math.com/library/slideshow/net-octahedron

Example 1 Tetrahedron

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

1. Watch the animation of a tetrahedron’s net folding into a 3D shape.
2. Cut out and fold the printable net to create a tetrahedron. 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 tetrahedron has 4 triangular faces.
• Edges: A tetrahedron has 6 edges.
• Vertices: A tetrahedron has 4 vertices.

Think About It: How does the tetrahedron compare to a cube in terms of stability?

Example 2 Square Pyramid

Now, let's examine a square pyramid!

1. Watch the animation of a square pyramid’s net folding.
2. Cut out the printable net and fold it into a square pyramid. 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: 5 faces (1 square base, 4 triangular faces).
• Edges: 8 edges.
• Vertices: 5 vertices.

Think About It: Why do pyramids appear in ancient architecture?

Example 3 Octahedron

Finally, let's investigate the octahedron!

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

You should find:

• Faces: 8 triangular faces.
• Edges: 12 edges.
• Vertices: 6 vertices.

Think About It: How is an octahedron similar to two pyramids joined together?

Example 4 Surface Area of a Square Pyramid

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

A square pyramid has a square base and four triangular faces. To find the total surface area, we use the formula:

\[ SA = B + \frac{1}{2} P l \]

Where:

• \( B \) is the area of the square base.
• \( P \) is the perimeter of the base.
• \( l \) is the slant height.

Let's Try It!

Suppose a square pyramid has a base side length of 5 cm and a slant height of 7 cm. Find the surface area.

\[ SA = (5 \times 5) + \frac{1}{2} (4 \times 5) (7) \] \[ SA = 25 + \frac{1}{2} (20 \times 7) \] \[ SA = 25 + \frac{1}{2} (140) \] \[ SA = 25 + 70 \] \[ SA = 95 \text{ cm}^2 \]

Final Answer: The surface area of this square pyramid is 95 cm².

Think About It: What happens to the surface area if the slant height is doubled?

Example 5 Surface Area of a Tetrahedron

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

A regular tetrahedron has four equilateral triangle faces. The surface area formula is:

\[ SA = \sqrt{3} e^2 \]

Where:

• \( e \) is the length of one edge.

Let's Try It!

Suppose a tetrahedron has an edge length of 6 cm. Find its surface area.

\[ SA = \sqrt{3} (6^2) \] \[ SA = \sqrt{3} (36) \] \[ SA \approx 62.35 \text{ cm}^2 \]

Final Answer: The surface area of this tetrahedron is approximately 62.35 cm².

Think About It: How does the surface area change if the edge length is doubled?

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.
• Pyramids and tetrahedra have triangular faces that provide structural stability.
• Nets are used in real-world applications, such as packaging, architecture, and bridge design.

Example 1 Egyptian Pyramids

Think about the famous Egyptian pyramids. They have stood for thousands of years because of their structural stability.

Let’s explore!

1. Research why pyramids are one of the strongest architectural structures.
2. What materials were used in building ancient pyramids?
3. How does the shape of a pyramid help distribute weight?

Think About It: How do modern engineers use the pyramid shape in construction today?

Example 2 Surface Area of an Octahedron

An octahedron consists of 8 triangular faces. To find its surface area, use the formula:

\[ SA = 2 \times \text{area of one triangular face} \times 4 \]

Let's Try It!

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

\[ SA = 2 \times \left( \frac{1}{2} (6 \times 5) \right) \times 4 \] \[ SA = 2 \times (15) \times 4 \] \[ SA = 120 \text{ cm}^2 \]

Final Answer: The surface area of the octahedron is 120 cm².

Think About It: Why are octahedral shapes used in molecular structures and geodesic domes?

Quiz

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

1. What is a net in geometry?
2. How many faces does a tetrahedron have?
3. How many edges does a square pyramid have?
4. How many faces, edges, and vertices does an octahedron have?
5. Calculate the surface area of a square pyramid with a base side length of 6 cm and a slant height of 8 cm.
6. Calculate the surface area of a tetrahedron with an edge length of 5 cm.
7. Why are pyramids considered one of the strongest structural designs?
8. How do engineers use tetrahedral structures in modern construction?
9. What would happen if a pyramid net were missing one of its triangular faces?
10. How can you determine if a net will fold into a correct 3D shape?

Answer Key

1. A net is a 2D pattern that folds into a 3D shape.
2. 4 faces.
3. 8 edges.
4. 8 faces, 12 edges, 6 vertices.
5. 144 cm².
6. 43.3 cm².
7. Triangular faces distribute weight evenly, making pyramids highly stable.
8. Tetrahedral structures are used in bridges, trusses, and strong frameworks.
9. The pyramid would be incomplete and unable to fully enclose space.
10. The net must have the correct number and arrangement of faces to create a closed 3D shape.