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32nd International Conference on Technology in Collegiate Mathematics

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VIRTUAL CONFERENCE

32nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE

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Exploring Volumes with GeoGebra Dissection Models

• Dr. Thomas (Tom) Cooper

Professor, University of North Georgia Department of Mathematics Dahlonega Campus tom.cooper@ung.edu http://faculty.ung.edu/tecooper/

32nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE

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BIG IDEA

• Software such as Geogebra allows us

to build, view, and manipulate 3D

• bjects.
• We can explore volume formulas

intuitively.

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Two Interesting Solids

Frustum of a Square Pyramid Burr Buzzle & Pieces

GeoGebra Model

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https://www.gathering4gardner.org/g4g13gift/math/ BanchoffThomas-GiftExchange-Foxtrot-G4G13.pdf

32nd International Conference on Technology in Collegiate Mathematics VIRTUAL CONFERENCE

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Both can be decomposed into square pyramids and “semi-orthocentric” tetrahedra.

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Some Volumes We Can Explore Intuitively

Volume of any Prism = height × (area of the base) Volume of any Pyramid =

𝟐 𝟒 × height × (area of the base)

= 𝟐 𝟒 × (volume of the prism it fits in)

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Some Volumes We Can Explore Intuitively

Volume of any tetrahedron

=

𝟐 𝟕 × (volume of the parallelepiped it fits in)

Volume of a “semi-orthocentric” tetrahedron with a pair of perpendicular opposite edges = 𝟐 𝟕 × "height" × (𝐪𝐬𝐩𝐞𝐯𝐝𝐮 𝐩𝐠 𝐮𝐢𝐩𝐭𝐟 𝐟𝐞𝐡𝐟 𝐦𝐟𝐨𝐡𝐮𝐢𝐭)

Now to Geogebra …

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The volume of a right rectangular prism is

LWH = height × (area of the base).

GeoGebra Model of Prisms

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• By Cavalieri’s Principle, the volume of

any prism is height × (area of the base).

GeoGebra Model of Cavalieri for Pyramids Geogebra Model of Cavalieri for Prisms

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Pyramids (Volume = 1/3 Prism)

• Geometric Argument by decomposition
• Any pyramid (including cone) by Cavalieri’s Principle

GeoGebra Model for a Cube More General Case (GGB) using a pyramid with a right triangle base

These were called “Yangma” by Liu Hui in The Nine Chapters

• n the

Mathematical Art , 263 AD

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Tetrahedron

Volume=

1 6 𝒃 × 𝒄 ∙ 𝒅

GeoGebra Construction

• By decomposition, Volume of tetrahedron

= 1/3 x (Volume of Triangular Prism) = 1/6 × (Volume of Parallelepiped)

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• When a pair of opposite edges are “perpendicular”, Volume

= 1/6 × h × product of these side lengths.

,0,0 , , , , , , a x y h x y b h = = = + a b c

(0) ( ) ( ) 0, , 1 1 0, , , , 6 6 1 0 ( )( ) 6 1 6 a x y h ah ay ah ay ah ay x y b h ah y b ayh abh  = = − + = −   = −  + = + − + + = i j k a b i j k a b c

Volume=

1 6 𝒃 × 𝒄 ∙ 𝒅

Geogebra Model

Semi-Orthocentric Tetrahedron

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Frustums (with Square Bases)

𝑊 = 𝑐2ℎ + 1 3 ℎ(𝑏 − 𝑐)2+4 × 1 2 𝑏 − 𝑐 2 𝑐ℎ = 𝑐2ℎ +

1 3 ℎ𝑏2 − 2 3 ℎ𝑏𝑐 + 1 3 ℎ𝑐2 + 𝑏𝑐ℎ − 𝑐2ℎ

=

1 3 ℎ𝑏2 + 1 3 ℎ𝑏𝑐 + 1 3 ℎ𝑐2

=

1 3 ℎ(𝑏2+𝑏𝑐 + 𝑐2)

GeoGebra Model

Liu Hui’s (263 AD)decomposition into a square prism, 4 “qiandus” (triangular prisms), and 4 “yangmas” (square pyramids)

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Dissection Idea of Tom Banchoff

(Professor Emeritus Brown University)

Divide the bottom into n by n squares, and divide the top into (n+1) by (n+1) squares. The frustum can be divided in: n2 pyramids pointing up, each with V =

1 3 ℎ 𝑏 𝑜 2

(n+1)2 pyramids pointing down, each with 𝑊 =

1 3 ℎ 𝑐 𝑜+1 2

2n(n+1)semi-orthocentric tetrahedra, with 𝑊 =

1 6 ℎ 𝑏 𝑜 𝑐 𝑜+1

This give a total volume of 𝑊 = 𝑜2 1 3 ℎ 𝑏 𝑜

2

+ 2𝑜 𝑜 + 1 1 6 ℎ 𝑏 𝑜 𝑐 𝑜 + 1 + (𝑜 + 1)2 1 3 ℎ 𝑐 𝑜 + 1

2

= 1 3 ℎ𝑏2 + 1 3 ℎ𝑏𝑐 + 1 3 ℎ𝑐2 = 1 3 ℎ 𝑏2 + 𝑏𝑐 + 𝑐2

GeoGebra Model

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Four Piece Dissection

GeoGebra Model

Consider the frustum with the top being an 𝑏 × 𝑏 square and the bottom being a 𝑐 × 𝑐 square. This can be dissected into pyramids with volumes 𝑊

1 = 1 3 ℎ𝑏2 and 𝑊 2 = 1 3 ℎ𝑐2 and semi-orthocentric tetrahedra with volumes

𝑊

3 = 1 6 ℎ𝑏𝑐 and 𝑊 4 = 1 6 ℎ𝑏𝑐.

So the frustum has volume 𝑊 =

1 3 ℎ 𝑏2 + 𝑏𝑐 + 𝑐2

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Burr Puzzle

2 2 3

1 1 2 Pyramids, each with: = 3 3 2 6 h s V s s s   = =    

2 2 3

1 1 1 4 Tetrahedra, each with: 6 6 2 12 s V hs s s   = = =    

3 3 3

2 4 2 Puzzle Piece Volume: 6 12 3 V s s s = + =

3 3

2 Total Volume: 6 4 3 V s s   = =     Note the total volume is half the volume

• f a cube with sides of length 2s.

GeoGebra Model

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Yoshimoto Cube

GeoGebra Demonstration

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Contact Information

• Dr. Tom Cooper

Professor University of North Georgia tecooper@ung.edu http://faculty.ung.edu/tecooper