Area, Surface Area, Volume, Cross Section, 3D

Tim Brzezinski, Sep 30, 2016

Table of Contents

  1. Perimeter, Area, Volume
    1. Unit Squares: Multipurpose Template
    2. Multipurpose Composite Figure Templates
    3. Open Middle: Maximizing and Minimizing Area (1)
    4. Composite Figure Template
    5. Open Middle: Perimeter of a Rectangle
    6. TRUE MEANING of π
    7. Circumference = ? (Animation)
    8. Animation 66
    9. Circumference = ? (Animation II)
    10. Animation 70
    11. Animation 65
    12. Parallelograms: Counting Squares
    13. Parallelograms: Area Exploration Templates
    14. Parallelogram: Area
    15. Building Parallelograms with Given Area: Quick Formative Assessment
    16. Animation 202
    17. Area of a Rhombus
    18. Area of a Triangle: Intuitive Discovery
    19. Triangle Area Action! (V1)
    20. Triangle Area Action!!! (V2)
    21. Area of a Triangle (V3)
    22. Animation 175
    23. Create a Triangle with Given Area: Quick Formative Assessment
    24. Create a Triangle with Given Area (V2)
    25. Open Middle: Triangle Area Problem (V2)
    26. Open Middle: Area of a House
    27. Area of a Trapezoid: Intuitive Discovery
    28. Area of a Trapezoid (3)
    29. Open Middle: Trapezoid Area Problem
    30. Open Middle Trapezoid Area Problem: Choose Your Own Fixed Perimeter
    31. Create a Trapezoid with Given Area: Quick Formative Assessment
    32. Area of a Circle
    33. Circle Area (by Peeling)
    34. Circle Area (By Peeling!)
    35. Area of a Circle - without words (Animation 38)
    36. Area of a Circle (III)
    37. Area of a Circle (by Peeling)
    38. Area of a Circle II
    39. Classic Cow Grazing Problem
    40. Build a Circle: Radius, Diameter, Circumference, Area
    41. Segment Area: Quick Investigation
    42. Cavalieri's Principle (祖暅原理)
    43. Cavalieri's Principle
    44. Figure 6.7 Cavalieri's principle
    45. Mailbox? Backpack? (Composite Solid)
    46. Washer Action!!! (1)
    47. Animation 169
    48. Surface Area: Introductory Exercises
    49. Open Middle: Maximizing Rectangular Prism Surface Area
    50. Maximizing Surface Area: Open Middle Cube Problem (V1)
    51. Maximizing Surface Area: Open Middle Cube Problem (V2)
    52. Open Middle (Revamped): Creating Rectangular Prisms with a Given Surface Area
    53. Trisecting the Cube into 3 Pyramids
    54. Volume of Pyramids
    55. Volume of Pyramids - Chinese Proof
  2. More with volume
    1. Unwrapping a Cylinder
    2. Unwrapping a Cylinder: REVAMPED!
    3. Volume of Spheres
    4. Volume of Spheres with Proof
    5. A sphere in a cylinder.
    6. Surface Area of Spheres
    7. Volumes and Surface Areas of Similar Cuboids
    8. Square Pyramid: Underlying Anatomy
    9. Cone Anatomy
    10. oblique and right pyramid - Cavalieri
    11. Build Your Own Right Triangular Prism (V2)!
    12. Build Your Own Right Triangular Prism!
    13. Right Triangular Prism!
    14. Adjustable Triangular Prism
    15. Sphere Peeling: Volume
    16. Net of a Cone
    17. Curved Surface Area of Cones (Combined Version)
    18. Volume of a Rectangular Prism
  3. 3D Coordinates & Modeling
    1. Plotting Points in 3D: Dynamic Illustrator
    2. Plotting Points in 3 Space
    3. Writing Ordered Triples
    4. Working in 3D: Warm Up Questions (1)
    5. Toblerone Modeling (Part 1)
    6. Toblerone Modeling (Part 2)
    7. 3D Modeling Using Coordinates (1): Building Rectangular Prisms
    8. 3D Modeling Using Coordinates (2): Building Composite Solids
    9. 3D Modeling Using Coordinates (3): Building Composite Solids
    10. 3D Modeling Using Coordinates (4): Building Composite Solids
  4. 2D to 3D and Cross Section
    1. Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR
    2. Surface of Revolution Template
    3. Surface of Revolution: GeoGebra Augmented Reality Template
    4. Plane Bisector: Quick Illustrator
    5. Sphere + Plane = ???
    6. Sphere + Plane = ?
    7. Sections of Rectangular Prisms (Cuboids)
    8. Sections of Triangular Prisms
    9. Sections of Cylinders
    10. Sections of Rectangular Pyramids
    11. Sections of Triangular Pyramids
    12. Sections of Cones
    13. Sections of Spheres
    14. Exploring Sections of Cubes
    15. 11 Nets of the Cube
    16. Cube Geometry: Shortest Path Between Points - A Net
    17. Cube Dissection Problem
    18. Rotating a Triangle Around a Coordinate Axis
    19. Box with Open Top
    20. Cross Section? (1)
    21. Surface with Square Cross Sections
  5. 3D: Surfaces of Revolution
    1. Cross Section to Surface of Revolution
    2. 2D to 3D: What's Going On?
    3. Creating Surfaces of Revolution
    4. Open Middle: Linear Function to Cone
    5. Warm Up: Creating Surfaces of Revolution (1)
    6. Warm Up: Creating Surfaces of Revolution (2)
    7. Warm Up: Creating Surfaces of Revolution (3)
    8. Cup Modeling Project: Building Surfaces of Revolution
  6. Other Geometry Resources
    1. Geometry Resources

1.

Perimeter, Area, Volume

  • 1. Unit Squares: Multipurpose Template
  • 2. Multipurpose Composite Figure Templates
  • 3. Open Middle: Maximizing and Minimizing Area (1)
  • 4. Composite Figure Template
  • 5. Open Middle: Perimeter of a Rectangle
  • 6. TRUE MEANING of π
  • 7. Circumference = ? (Animation)
  • 8. Animation 66
  • 9. Circumference = ? (Animation II)
  • 10. Animation 70
  • 11. Animation 65
  • 12. Parallelograms: Counting Squares
  • 13. Parallelograms: Area Exploration Templates
  • 14. Parallelogram: Area
  • 15. Building Parallelograms with Given Area: Quick Formative Assessment
  • 16. Animation 202
  • 17. Area of a Rhombus
  • 18. Area of a Triangle: Intuitive Discovery
  • 19. Triangle Area Action! (V1)
  • 20. Triangle Area Action!!! (V2)
  • 21. Area of a Triangle (V3)
  • 22. Animation 175
  • 23. Create a Triangle with Given Area: Quick Formative Assessment
  • 24. Create a Triangle with Given Area (V2)
  • 25. Open Middle: Triangle Area Problem (V2)
  • 26. Open Middle: Area of a House
  • 27. Area of a Trapezoid: Intuitive Discovery
  • 28. Area of a Trapezoid (3)
  • 29. Open Middle: Trapezoid Area Problem
  • 30. Open Middle Trapezoid Area Problem: Choose Your Own Fixed Perimeter
  • 31. Create a Trapezoid with Given Area: Quick Formative Assessment
  • 32. Area of a Circle
  • 33. Circle Area (by Peeling)
  • 34. Circle Area (By Peeling!)
  • 35. Area of a Circle - without words (Animation 38)
  • 36. Area of a Circle (III)
  • 37. Area of a Circle (by Peeling)
  • 38. Area of a Circle II
  • 39. Classic Cow Grazing Problem
  • 40. Build a Circle: Radius, Diameter, Circumference, Area
  • 41. Segment Area: Quick Investigation
  • 42. Cavalieri's Principle (祖暅原理)
  • 43. Cavalieri's Principle
  • 44. Figure 6.7 Cavalieri's principle
  • 45. Mailbox? Backpack? (Composite Solid)
  • 46. Washer Action!!! (1)
  • 47. Animation 169
  • 48. Surface Area: Introductory Exercises
  • 49. Open Middle: Maximizing Rectangular Prism Surface Area
  • 50. Maximizing Surface Area: Open Middle Cube Problem (V1)
  • 51. Maximizing Surface Area: Open Middle Cube Problem (V2)
  • 52. Open Middle (Revamped): Creating Rectangular Prisms with a Given Surface Area
  • 53. Trisecting the Cube into 3 Pyramids
  • 54. Volume of Pyramids
  • 55. Volume of Pyramids - Chinese Proof

1.1.

Unit Squares: Multipurpose Template

Choose any number of squares from 1-60. You can move these squares around.
Choose any number of squares from 1-60. You can move these squares around.
Choose any number of squares from 1-60. You can move these squares around.
Choose any number of squares from 1-60. You can move these squares around.
– Tim Brzezinski

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2.

More with volume

  • 1. Unwrapping a Cylinder
  • 2. Unwrapping a Cylinder: REVAMPED!
  • 3. Volume of Spheres
  • 4. Volume of Spheres with Proof
  • 5. A sphere in a cylinder.
  • 6. Surface Area of Spheres
  • 7. Volumes and Surface Areas of Similar Cuboids
  • 8. Square Pyramid: Underlying Anatomy
  • 9. Cone Anatomy
  • 10. oblique and right pyramid - Cavalieri
  • 11. Build Your Own Right Triangular Prism (V2)!
  • 12. Build Your Own Right Triangular Prism!
  • 13. Right Triangular Prism!
  • 14. Adjustable Triangular Prism
  • 15. Sphere Peeling: Volume
  • 16. Net of a Cone
  • 17. Curved Surface Area of Cones (Combined Version)
  • 18. Volume of a Rectangular Prism

2.1.

Unwrapping a Cylinder

– Tim Brzezinski

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3.

3D Coordinates & Modeling

  • 1. Plotting Points in 3D: Dynamic Illustrator
  • 2. Plotting Points in 3 Space
  • 3. Writing Ordered Triples
  • 4. Working in 3D: Warm Up Questions (1)
  • 5. Toblerone Modeling (Part 1)
  • 6. Toblerone Modeling (Part 2)
  • 7. 3D Modeling Using Coordinates (1): Building Rectangular Prisms
  • 8. 3D Modeling Using Coordinates (2): Building Composite Solids
  • 9. 3D Modeling Using Coordinates (3): Building Composite Solids
  • 10. 3D Modeling Using Coordinates (4): Building Composite Solids

3.1.

Plotting Points in 3D: Dynamic Illustrator

[size=150]Location in space is represented using an ordered triple: [math]\left(x,y,z\right)[/math]. [br]This app illustrates how we plot various points in 3-space. [/size]
To move the point up and down (parallel to the zAxis), click on it first before dragging it up and down. Click it again to move it around at the same level above or below the grid.
– Tim Brzezinski

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4.

2D to 3D and Cross Section

  • 1. Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR
  • 2. Surface of Revolution Template
  • 3. Surface of Revolution: GeoGebra Augmented Reality Template
  • 4. Plane Bisector: Quick Illustrator
  • 5. Sphere + Plane = ???
  • 6. Sphere + Plane = ?
  • 7. Sections of Rectangular Prisms (Cuboids)
  • 8. Sections of Triangular Prisms
  • 9. Sections of Cylinders
  • 10. Sections of Rectangular Pyramids
  • 11. Sections of Triangular Pyramids
  • 12. Sections of Cones
  • 13. Sections of Spheres
  • 14. Exploring Sections of Cubes
  • 15. 11 Nets of the Cube
  • 16. Cube Geometry: Shortest Path Between Points - A Net
  • 17. Cube Dissection Problem
  • 18. Rotating a Triangle Around a Coordinate Axis
  • 19. Box with Open Top
  • 20. Cross Section? (1)
  • 21. Surface with Square Cross Sections

4.1.

Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR

Students & Teachers with ARCore by Google Installed on Your Devices:
Here's how you can create a 3D surface by rotating the graph of ANY function about the xAxis within [url=https://play.google.com/store/apps/details?id=org.geogebra.android.g3d]GeoGebra's 3D Graphing Calculator app[/url]. (This screencast was recorded on an ACER Chrome Tab 10.) [br][br]For more information re: getting started with GeoGebra Augmented Reality, [url=https://www.geogebra.org/m/jhcywuqw]click here[/url].
Quick Demo
Now, press the [b]AR [/b]button (lower right) to explore this within GeoGebra Augmented Reality! [br][br]Once viewing in Augmented Reality, slide the slider slowly. You should see a surface of revolution form!
Here's what this surface looks like in GeoGebra Augmented Reality!
For further exploration:
[b]Note[/b]:[br][br]If interested, you can use this pre-made template [url=https://www.geogebra.org/m/qbxbcmqw#material/yt22anmh]that can be found here[/url]. [br]Follow the directions on this page to open in the [color=#1e84cc]GeoGebra 3D Graphing Calculator[/color] on your device. [br][br]1) Instead of f(x) = 3, try typing [math]f\left(x\right)=x,0\le x\le4[/math] (just the way you see it). Spin this graph about the [br] xAxis. What kind of solid do you get? [br][br]2) Same direction as in (1), but do this time for [math]f\left(x\right)=x,2\le x\le4[/math]. What do you get now? [br][br]3) Same direction as in (1), but do this time for [math]f\left(x\right)=0.5x^2,0\le x\le2[/math] . What does this look like? [br][br]4) What other kinds of 3D surfaces of revolution can we create this way? [br] Let your creativity take you places!
Another demo: Modeling a bowl
– Tim Brzezinski

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5.

3D: Surfaces of Revolution

  • 1. Cross Section to Surface of Revolution
  • 2. 2D to 3D: What's Going On?
  • 3. Creating Surfaces of Revolution
  • 4. Open Middle: Linear Function to Cone
  • 5. Warm Up: Creating Surfaces of Revolution (1)
  • 6. Warm Up: Creating Surfaces of Revolution (2)
  • 7. Warm Up: Creating Surfaces of Revolution (3)
  • 8. Cup Modeling Project: Building Surfaces of Revolution

5.1.

Cross Section to Surface of Revolution

[b][size=150][url=https://www.geogebra.org/m/jmtxqwyq]Click here to access the full activity[/url].[/size][/b]
– Tim Brzezinski

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6.

Other Geometry Resources

  • 1. Geometry Resources

6.1.

Geometry Resources

[list][*][b][url=https://www.geogebra.org/m/z8nvD94T]Congruence (Volume 1)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/munhXmzx]Congruence (Volume 2)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/dPqv8ACE]Similarity, Right Triangles, Trigonometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/C7dutQHh]Circles[/url][/b][/*][*][b][url=https://www.geogebra.org/m/K2YbdFk8]Coordinate and Analytic Geometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/xDNjSjEK]Area, Surface Area, Volume, 3D, Cross Section[/url] [/b][/*][*][b][url=https://www.geogebra.org/m/NjmEPs3t]Proof Challenges[/url]  [/b][/*][/list]
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
– Tim Brzezinski

Information