Directed Activity: The Concept of Volume

Shape

Surface Area and Volume--Directed Activity: The Concept of Volume

Focus

The meaning of volume.

Technologies

n/a

Background

Figure 6.10: Generating a 3-Dimensional Box from a 2-Dimensional Square

In Figure 6.10, a filled-in 2-dimensional object is used to generate a filled-in 3-dimensional object. As filled-in square ABCD moves in a direction perpendicular to itself, its successive locations determine a filled-in box.

Tasks

1. Does a square have volume? Why or why not?

2. Does a box have area? If so, in what sense?

3. How is the concept of volume related to the concept of area?

4. Does a shadow on a desktop have volume? Why or why not?

5. Could a 2-dimensional object have mass? Weight? Why or why not?

6. Does a sheet of paper have volume? Why or why not? If yes, how would you determine its volume?

7. What name would you give to a 3-dimensional object generated by moving a filled-in circle (i.e., a disc) in a direction perpendicular to itself?

8. What name would you give to a 3-dimensional object generated by moving a filled-in equilateral triangle in a direction perpendicular to itself?

9. The box in Figure 6.10 and the objects described in #3 and #4 have different shapes. They also have volume. Is the concept of volume in any way different when applied to objects with different shapes, or does it always mean the same thing?

10. How would you compute the volume of each object in #6?

11. How would you measure the volume of each object in #6?

12. How would you try to explain the concept of volume to an inhabitant of Flatland?

13. If you were an inhabitant of Flatland, and someone or something tried to explain the concept of volume to you, what conceptual difficulties might you encounter?