In the previous part, you learned that the relationship between perimeter and area is dynamic; namely, the amount of area or perimeter of a shape is not fixed in relation to the measure of the other variable.
How about the relationship between surface area and volume? Do prisms with the same volume have the same surface area? Let's explore this relationship.
Take 24 multilink cubes or building blocks and imagine that each cube represents a fancy chocolate truffle. For shipping purposes, these truffles need to be packaged into boxes in the shape of rectangular prisms. Knowing that you must always package 24 truffles (i.e., your volume is set at 24 cubic units), what are the possible dimensions for the boxes? Record your information in the table below:
The dimensions must all be factors of 24 -- 1, 2, 3, 4, 6, 8, 12, 24. Close Tip
Which of your packaging arrangements requires a box with the least amount of material? The greatest amount of material? Why is the amount of material needed for packaging important?
Problem B3
What do you notice about the shape of the package that has the smallest surface area? How about the package with the greatest surface area?
When the volume is constant (as in the truffles problem), the surface area depends on the shape of the solid. But what happens to the surface area of a solid as its volume increases -- does surface area increase at the same rate as volume?
Problem B4
Use multilink cubes or building blocks to create different-sized cubes from the table below. Calculate the volumes and surface areas of the cubes. Examine the proportional relationship between surface area (SA) and volume (V) by creating a surface area-to-volume ratio (SA:V). As volume increases, what happens to the ratio of surface area to volume?
You may find it helpful to build the different-sized cubes first and then to use the models to confirm your calculations of the volumes and surface areas. If cubes aren't available, you could make a sketch of the cubes on graph paper to help you visualize the surface area of each face. Write the surface area-to-volume ratios and look for patterns. Close Tip
Size of Cube
Surface Area
Volume
Ratio SA:V
1 by 1 by 1
6 • 12 = 6
13 = 1
6:1
2 by 2 by 2
6 • 22 = 24
23 = 8
6:2 (or 3:1)
3 by 3 by 3
6 • 32 = 54
33 = 27
6:3 (or 2:1)
4 by 4 by 4
6 • 42 = 96
43 = 64
6:4 (or 3:2)
5 by 5 by 5
6 • 52 = 150
53 = 125
6:5
6 by 6 by 6
6 • 62 = 216
63 = 216
6:6 (or 1:1)
7 by 7 by 7
6 • 72 = 294
73 = 343
6:7
8 by 8 by 8
6 • 82 = 384
83 = 512
6:8 (or 3:4)
In general, the ratio is 6:s, since the surface area formula is 6s2 and the volume formula is s3.
Here is the graph that shows what happens to the ratio of surface area to volume as the volume increases:
Video Segment Watch this video segment after you've completed Problem B4 to see what Lori and Jayne found out about the relationship between the volume and surface area of solid objects.
Did you express this relationship differently?
If you are using a VCR, you can find this segment on the session video approximately 11 minutes and 49 seconds after the Annenberg Media logo.
Problem B5
Over the last decade, we have seen the genesis of the giant "superstore." What kind of surface area-to-volume ratio do you think superstores have? Why do companies build such large stores?
