Area, Surface Area, Volume

Duke, Feb 4, 2022

Area, Surface Area, Volume, 6-8 Geometry, Horry

Table of Contents

  1. Area & Circumference
    1. Area of a Rectangle
    2. Area of a Triangle (Discovery)
    3. Parallelogram: Area
    4. Area of a Trapezoid
    5. Circumference and Pi
    6. Tennis Ball Can
    7. Rolling Quarter Puzzle
    8. Area of Circles
    9. Circle Area (By Peeling!)
    10. Area of a composite shape
  2. Surface Area & Nets
    1. Surface Area: Introductory Exercises
    2. Rectangular Prism: Basic Net Demo
    3. Surface area of rectangular prisms
    4. Square Pyramid: Underlying Anatomy
    5. Net of a Square Pyramid
    6. Net and Surface Area of Triangular Prism
    7. Build Your Own Right Triangular Prism (V2)!
    8. Net of Hexagonal Prism
    9. Net of a Cylinder
    10. Unwrapping a Cylinder: REVAMPED!
    11. Cone Anatomy
    12. Net of a Cone
    13. Curved Surface Area of Cones
    14. Surface Area of Spheres
  3. Cross-Sections
    1. Cube Cross Sections
    2. Cross Section of a cube
    3. Sections of Rectangular Prisms (Cuboids)
    4. Cross Section of a cuboid
    5. Sections of Triangular Prisms
    6. Cross Section of a pyramid
    7. Sections of Rectangular Pyramids
    8. Sections of Triangular Pyramids
    9. Sections of Cylinders
    10. Cross Section of a cone
    11. Sections of Cones
    12. Cross Section of a sphere
    13. Sections of Spheres
    14. Cavalieri's Principle
  4. Volume
    1. Volume: Intuitive Introduction
    2. Trisecting the Cube into 3 Pyramids
    3. Square Pyramid: Underlying Anatomy
    4. Volume of cylinder
    5. Cone Anatomy
    6. Volume Cylinder Sphere Cone
    7. The Volume of the Sphere
    8. Volume of Spheres
    9. Sphere Peeling: Volume
    10. Mailbox? Backpack? (Composite Solid)
    11. 3D Problem Solving - Sphere resting inside of a Cone
  5. Angle Relationships
    1. Vertical Angles Theorem
    2. Angle Relationships Problem (7.G.2.5)
    3. Transversal Intersects Parallel Lines
    4. Triangle Angle Theorems
    5. Triangle Exterior Angle
  6. Pythagorean's Theorem
    1. Visual Proof of Pythagorean Theorem
    2. Pythagorean Theorem Visual
    3. Pythagorean Jigsaw

1.

Area & Circumference

  • 1. Area of a Rectangle
  • 2. Area of a Triangle (Discovery)
  • 3. Parallelogram: Area
  • 4. Area of a Trapezoid
  • 5. Circumference and Pi
  • 6. Tennis Ball Can
  • 7. Rolling Quarter Puzzle
  • 8. Area of Circles
  • 9. Circle Area (By Peeling!)
  • 10. Area of a composite shape

1.1.

Area of a Rectangle

In this worksheet you will investigate the area of a rectangle.
Investigation:[br][br]1) Move the sliders Base and Height and describe what happens to the image.[br][br] 2) What relationship does the number of squares have with the area of the rectangle?[br][br] 3) Is there a way to find the number of square with out counting them?[br][br] 4) What conjecture can you make about the formula for area of a rectangle?
– GeoGebra Materials Team

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2.

Surface Area & Nets

  • 1. Surface Area: Introductory Exercises
  • 2. Rectangular Prism: Basic Net Demo
  • 3. Surface area of rectangular prisms
  • 4. Square Pyramid: Underlying Anatomy
  • 5. Net of a Square Pyramid
  • 6. Net and Surface Area of Triangular Prism
  • 7. Build Your Own Right Triangular Prism (V2)!
  • 8. Net of Hexagonal Prism
  • 9. Net of a Cylinder
  • 10. Unwrapping a Cylinder: REVAMPED!
  • 11. Cone Anatomy
  • 12. Net of a Cone
  • 13. Curved Surface Area of Cones
  • 14. Surface Area of Spheres

2.1.

Surface Area: Introductory Exercises

Take a few minutes to interact with the rectangular prism shown here. After doing so, create one that has a length = 4 units, width = 5 units, and height = 3 units.
How many square units (i.e. "squares") appear on 1 [b][color=#ff00ff]pink face[/color][/b]? [br]How many square units (i.e. "squares") appear on 1 [b][color=#bf9000]gold face[/color][/b]? [br]How many square units (i.e. "squares") appear on 1 [b]white face[/b]?
Use your answers for (1) to determine the TOTAL SURFACE AREA of this rectangular prism. [br]That is, how many square units, or squares, cover the ENTIRE SURFACE of this rectangular prism?
Now create a rectangular prism that has a length = 8 units, width = 3 units, and height = 5 units.
How many square units (i.e. "squares") appear on 1 [b][color=#ff00ff]pink face[/color][/b]? [br]How many square units (i.e. "squares") appear on 1 [b][color=#bf9000]gold face[/color][/b]? [br]How many square units (i.e. "squares") appear on 1 [b]white face[/b]?
Use your answers for (3) to determine the TOTAL SURFACE AREA of this rectangular prism. [br]That is, how many square units, or squares, cover the ENTIRE SURFACE of this rectangular prism?
Overall, how we can determine the total surface area (number of squares) that cover the entire surface of a rectangular prism? Describe.
– Tim Brzezinski

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3.

Cross-Sections

  • 1. Cube Cross Sections
  • 2. Cross Section of a cube
  • 3. Sections of Rectangular Prisms (Cuboids)
  • 4. Cross Section of a cuboid
  • 5. Sections of Triangular Prisms
  • 6. Cross Section of a pyramid
  • 7. Sections of Rectangular Pyramids
  • 8. Sections of Triangular Pyramids
  • 9. Sections of Cylinders
  • 10. Cross Section of a cone
  • 11. Sections of Cones
  • 12. Cross Section of a sphere
  • 13. Sections of Spheres
  • 14. Cavalieri's Principle

3.1.

Cube Cross Sections

Move the points I, K, and J around on the edges of the cube.
1) What are the different cross section shapes that you can make?[br]2) What is the cross section with the least number of sides?[br]3) What is the cross section with the most number of sides?[br]4) Can you make any regular polygonal cross sections? How would you position the points for each?
– Edward Knote

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4.

Volume

  • 1. Volume: Intuitive Introduction
  • 2. Trisecting the Cube into 3 Pyramids
  • 3. Square Pyramid: Underlying Anatomy
  • 4. Volume of cylinder
  • 5. Cone Anatomy
  • 6. Volume Cylinder Sphere Cone
  • 7. The Volume of the Sphere
  • 8. Volume of Spheres
  • 9. Sphere Peeling: Volume
  • 10. Mailbox? Backpack? (Composite Solid)
  • 11. 3D Problem Solving - Sphere resting inside of a Cone

4.1.

Volume: Intuitive Introduction

STUDENTS:
Interact with the applet below for a few minutes. Then answer the questions that follow. [br][br][b][color=#1e84cc]To explore this resource in Augmented Reality, see the directions beneath the questions listed below. [/color][/b]
1.
In the applet above, create a rectangular prism that has its first layer measuring 3 units long by 4 units wide. Suppose this rectangular prism is 2 layers high. How many cubes make up this prism?
2.
In the applet above, create a rectangular prism that has its first layer measuring 5 units long by 3 units wide. How many cubes make up this prism if this prism is 2 layers high? 3 layers high? 10 layers high?
3.
How can we EASILY determine the number of cubes that fit inside ANY rectangular prism? Describe.
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the[b] MENU (horizontal bars)[/b] in the upper left corner. Select [b]OPEN[/b]. [br] In the Search GeoGebra Resources input box, type [b]dp6ghmvv[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br]3) In the resource that uploads, zoom out and/or adjust the [br] LENGTH, WIDTH, & HEIGHT sliders to create a prism with dimensions you like. [br][br] Press the [b]AR[/b] button in the lower right corner of your 3D screen. Follow the directions that appear.
– Tim Brzezinski

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5.

Angle Relationships

  • 1. Vertical Angles Theorem
  • 2. Angle Relationships Problem (7.G.2.5)
  • 3. Transversal Intersects Parallel Lines
  • 4. Triangle Angle Theorems
  • 5. Triangle Exterior Angle

5.1.

Vertical Angles Theorem

[b]Definition:[/b] [b][color=#b20ea8]Vertical Angles[/color][/b] are angles whose sides form 2 pairs of opposite rays. [br][br]When 2 lines intersect, 2 pairs of vertical angles are formed. [color=#b20ea8]One pair of vertical angles is shown below. [/color] [br][color=#888](Click the other checkbox on the right to display the other pair of vertical angles.) [/color][br][br]Interact with the following applet for a few minutes, then answer the questions that follow.
Directions & Questions: [br][br]1) Complete the following statement (based upon your observations). [br] [br] [color=#b20ea8][b]Vertical angles are always __________________________.[/b] [/color] [br][br]2) Suppose the pink angle above measures 140 degrees. What would the measure of its vertical angle? What would be the measure of the other 2 (gray) angles?
– Tim Brzezinski

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6.

Pythagorean's Theorem

  • 1. Visual Proof of Pythagorean Theorem
  • 2. Pythagorean Theorem Visual
  • 3. Pythagorean Jigsaw

6.1.

Visual Proof of Pythagorean Theorem

For a right triangle, the square of the hypotenuse length [b]c[/b] equals the sum of the squares [br]of the leg lengths [b]a[/b] and [b]b[/b].
The area [b]c[sup]2[/sup][/b] of the big square transforms into the sum[br]sum [b]a[sup]2[/sup] + b[sup]2[/sup][/b] of the areas of the two smaller squares.
Developed by Jennifer Talavage, who was inspired by Stephen Kent Stephenson.
– kosband, Jennifer Talavage

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