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Write-up


Title
Volume and Surface Area of A Cube

Problem Statement
Suppose you build a cube whose length, width, and height are each 2 cm. What is the volume of this cube? What is its surface area? Repeat this procedure for a 3 x 3 x 3 cube and then for a 4 x 4 x 4 cube. What patterns do you notice? Can you predict the volume and surface area of a 17 x 17 x 17 cube? How about an s x s x s cube?

Problem setup

I set up the problem using the Excel Spreadsheet. For Surface Area, I will use a formula derived from the definition of a rectangular prism.. For the Volume, I will use a formula also derived from the definition of a rectangular prism.

 

Plans to Solve/Investigate the Problem

For Surface Area, I used a formula which calculates the area of each of the 6 sides, and then adds them together. For the Volume, I used a formula which multiplied the Length times the Width times the Height. I enter the length of the sides, beginning with 3 and calculated up to 21.

 

Investigation/Exploration of the Problem

The patterns that emerged were as follows:

 

Length

Width

Heighth

Surface Area

Volume

2

2

2

24

8

3

3

3

54

27

4

4

4

96

64

5

5

5

150

125

6

6

6

216

216

7

7

7

294

343

8

8

8

384

512

9

9

9

486

729

10

10

10

600

1000

11

11

11

726

1331

12

12

12

864

1728

13

13

13

1014

2197

14

14

14

1176

2744

15

15

15

1350

3375

16

16

16

1536

4096

17

17

17

1734

4913

18

18

18

1944

5832

19

19

19

2166

6859

20

20

20

2400

8000

21

21

21

2646

9261

 

 

SA = 6 times s˛

V = sł

 

This is as I expected given our knowledge of squares and cubes. This also matches Dr. Math’s formulas for cubes.

 

Extensions of the Problem

The extensions that were given are very inviting, yet it was difficult to imagine how I would perform the volume displacement of a solid object in the computer lab. Perhaps I will try this at a different location, provided time and resources are available.

 

Find the volume of an irregularly shaped large object. First, find a small plastic model of that object - one that you wouldn't mind getting wet (like a model of a dinosaur). Next, record the volume of the model using displacement. That is, fill a graduated cylinder about half-way with water and record the volume; then, completely submerge the model into the water and record the new volume. The volume of the model can be determined by subtracting the original volume from the new volume. Finally, find a scale factor between the model and the large object, and then use this scale factor and the model's volume to compute the volume of the original object.

Author & Contact
Jim Taylor
jtaylor1@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
Cube / Rectangular Prism Activity
http://mathforum.org/alejandre/escot/cube.prism.html