Shapes 3D

Arthur Lee, Nov 28, 2014

Solids and 3D geometry

Table of Contents

  1. Solids
    1. 3D Pythagorean Theorem
    2. Folding Aeroplane (HKCEE 1999 )
    3. Coxeter's Kaleidoscopic Cell 1
    4. Spherical Lines: Great Circles
    5. Parabola Around y-Axis
    6. Cube and Tetrahedron
    7. cube in dodecahedron
  2. Symmetry
    1. Rotational Symmetry of Cube
    2. Rotational Symmetry of Tetrahedron
    3. Rotational Symmetry of Octahedron
    4. Rotational Symmetry of Icosahedron
    5. Rotational Symmetry of Dodecahedron
  3. Nets
    1. Nets of the Cube
    2. 11 Nets of the Cube
    3. Net of Hexagonal Prism
    4. Nets of Tetrahedron
    5. Net of Icosahedron
    6. Net of Dodecahedron
    7. Another Net of Dodecahedron
    8. Octahedron and Boat
    9. The Quadratic Pyramid
  4. Sections
    1. Exploring Sections of Cubes
    2. Sections of Rectangular Prisms (Cuboids)
    3. Sections of Triangular Prisms
    4. Sections of Rectangular Pyramids
    5. Sections of Triangular Pyramids
  5. Projection
    1. projection of a segment
    2. opening a door
    3. box diagonal
    4. Shadow
    5. Projection of a Sphere - Central Projection
  6. Surface
    1. Curved Surface Area of Cones (Parametric Curved Surface)
    2. Surface Area of Spheres
    3. Curved Surface Area of Cones
    4. Curved Surface Area of Cones (Combined Version)
    5. Making a cone
  7. Volume
    1. equal cross sections
    2. three tetrahedrons
    3. tisect a cube
    4. opening a cube
    5. cutting a sphere
    6. section of sphere
    7. section of cone
    8. Kugelvolumen - Cavalieri
    9. Volume of Spheres
    10. A sphere in a cylinder.
    11. maximum cone
  8. Dumpling models
    1. dumpling model
    2. Tetrahedron Net (general)
    3. tetrahedron unfold
    4. The Dumpling Challenge (inspired by Arthur Lee)
    5. The Dumpling Challenge II
    6. The Dumpling Challenge III
    7. The Final Dumpling Challenge

1.

Solids

  • 1. 3D Pythagorean Theorem
  • 2. Folding Aeroplane (HKCEE 1999 )
  • 3. Coxeter's Kaleidoscopic Cell 1
  • 4. Spherical Lines: Great Circles
  • 5. Parabola Around y-Axis
  • 6. Cube and Tetrahedron
  • 7. cube in dodecahedron

1.1.

3D Pythagorean Theorem

Here is a 3D version of the Pythagorean Theorem. [ABC] means Area of triangle ABC. Drag points B, C, and D.
3D Pythagorean Theorem
– Steve Phelps

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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2.

Symmetry

  • 1. Rotational Symmetry of Cube
  • 2. Rotational Symmetry of Tetrahedron
  • 3. Rotational Symmetry of Octahedron
  • 4. Rotational Symmetry of Icosahedron
  • 5. Rotational Symmetry of Dodecahedron

2.1.

Rotational Symmetry of Cube

Drag the red points on the top and the bottom of the cube to investigate its rotational symmetry.
– Anthony OR 柯志明

2.2.

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2.3.

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2.4.

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2.5.

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3.

Nets

  • 1. Nets of the Cube
  • 2. 11 Nets of the Cube
  • 3. Net of Hexagonal Prism
  • 4. Nets of Tetrahedron
  • 5. Net of Icosahedron
  • 6. Net of Dodecahedron
  • 7. Another Net of Dodecahedron
  • 8. Octahedron and Boat
  • 9. The Quadratic Pyramid

3.1.

Nets of the Cube

Anthony Or[br]GeoGebra Institute of Hong Kong.
– Anthony OR 柯志明

3.2.

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3.3.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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4.

Sections

  • 1. Exploring Sections of Cubes
  • 2. Sections of Rectangular Prisms (Cuboids)
  • 3. Sections of Triangular Prisms
  • 4. Sections of Rectangular Pyramids
  • 5. Sections of Triangular Pyramids

4.1.

Exploring Sections of Cubes

Drag the red points to see the different sections of the cube.
Anthony Or. GeoGebra Institute of Hong Kong
– Anthony OR 柯志明

4.2.

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4.5.

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5.

Projection

  • 1. projection of a segment
  • 2. opening a door
  • 3. box diagonal
  • 4. Shadow
  • 5. Projection of a Sphere - Central Projection

5.1.

projection of a segment

projection of a segment
– Arthur Lee

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5.5.

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6.

Surface

  • 1. Curved Surface Area of Cones (Parametric Curved Surface)
  • 2. Surface Area of Spheres
  • 3. Curved Surface Area of Cones
  • 4. Curved Surface Area of Cones (Combined Version)
  • 5. Making a cone

6.1.

Curved Surface Area of Cones (Parametric Curved Surface)

What is the curved surface area of the cone?[br]Press the red point in the figure.[br]See http://ggbtu.be/m161406 to find the curved surface area.
– Anthony OR 柯志明

6.2.

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6.3.

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6.5.

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7.

Volume

  • 1. equal cross sections
  • 2. three tetrahedrons
  • 3. tisect a cube
  • 4. opening a cube
  • 5. cutting a sphere
  • 6. section of sphere
  • 7. section of cone
  • 8. Kugelvolumen - Cavalieri
  • 9. Volume of Spheres
  • 10. A sphere in a cylinder.
  • 11. maximum cone

7.1.

equal cross sections

Two triangular pyramids of same base and height are cut by a horizontal plane. The apexes D and E can be dragged horizontally and independently. [br]Drag G to change the height of the cutting plane. The cross sections should be the same. [br]View from top to see more clearly the shape of the bases and sections.[br]According to the Cavalieri's Principle*, the volume of these pyramids should be equal.
equal cross sections
* [url]http://en.wikipedia.org/wiki/Cavalieri's_principle[/url]
– Arthur Lee

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7.6.

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7.7.

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7.8.

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7.9.

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7.10.

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7.11.

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8.

Dumpling models

  • 1. dumpling model
  • 2. Tetrahedron Net (general)
  • 3. tetrahedron unfold
  • 4. The Dumpling Challenge (inspired by Arthur Lee)
  • 5. The Dumpling Challenge II
  • 6. The Dumpling Challenge III
  • 7. The Final Dumpling Challenge

8.1.

dumpling model

This is a rice dumpling model in the shape of a tetrahedron with 4 faces of isosceles triangles. In the 3D model on the right, you may drag D, E and F to fold faces. [br][br]Imagine where the faces will meet. Make a guess and use the checkboxes to find out more about the shape.
If you drag C on the left, the 4 triangles will be changed but remains congruent. How would you describe, in general, which point on the plane will be the projection of the apex? [br][br]Check another version where the faces can be changed individually:[br]https://www.geogebra.org/m/hdgmkgsa [br][br]or this version, focusing on its apex and altitude:[br]https://www.geogebra.org/m/wdkbbq49
– Arthur Lee

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8.7.

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