3. Determine the surface area, volume, and surface area to volume ratio for the following three blocks

Question

0.5 cm x 0.5 cm x 6.0

Answer 1

Imagine a rug laying on the floor. It takes up a certain amount of area. To calculate you would measure one side, then the other, and then multiply the two numbers. For example: 3 feet wide times 5 feet long equals 15 square feet. Area equals one side multiplied by the other.

Apply this concept to the block above. You could imagine six different rugs, one for each side. So, to find the surface area of this box you would measure each rug and add them up. See below:

1. 6 surfaces for this block: Front, back, top, bottom, left, and right
2. Area for each surface is one side multiplied by the other side (cm x cm = cm²)
3. Examples (cm shown for front only):
4. Front = 0.5 cm x 6 cm = 3 cm²
5. Back = 0.5 x 6 = 3
6. Top = 0.5 x 6 = 3
7. Bottom = 0.5 x 6 = 3
8. Left = 0.5 x 0.5 = 0.25
9. Right = 0.5 x 0.5 = 0.25
10. Total = 3 + 3 + 3 + 3 + 0.25 + 0.25 = 12.5 cm²

Volume is different than surface area. Imagine the rug analogy above. If you laid one rug on the floor it would take up 15 square feet of surface, but it would only be one rug thick. Now begin stacking more and more rugs on top. The surface touching the floor would stay the same, just one rug, but the pile would be getting higher. You could say the stack of rugs is taking up a greater volume of space. To calculate this you would take the surface area, 15 square feet, and then multiply by the height. If the rugs were stacked 1 foot thick then the volume would be 15 CUBIC feet. If the rugs were stacked 10 feet thick, then the volume would be 150 cubic feet. In other words, volume is width x depth x height. See example below:

1. Block Width = 6 cm
2. Depth = 0.5 cm
3. Height = 0.5 cm
4. Volume = 0.5 cm x 0.5 cm x 6 cm = 1.5 cm³ (cubic cm)

Surface area to volume ratio is an important concept in living systems. One example is heat exchange. The example is best illustrated using simple numbers, so instead of rugs we will use 1 cm cubes. A 1 cm cube has 6 sides that are all 1 cm by 1 cm. So the surface area is 1 cm² times 6 sides equals 6 cm². The volume is just 1 cm³ (1 cm x 1 cm x 1 cm). In other words the ratio of SA to Volume (SA:V) would be 6:1.

Now make the cube larger by adding 1 cube to the length, width, and height. Now, instead of 1 x 1 x 1, it would be 2 x 2 x 2. Lets calculate the SA. One side would by 2 cm times 2 cm for an area of 4 cm². Multiply that by 6 sides and the surface area would be 6 x 4 cm² equals 24 cm². Volume however would be 2 cm x 2 cm x 2 cm, or 8 cm³. The SA:V ratio would then be 24:8.

Looking at these trends together you can generalize that volume and surface area don't increase at the same rate. Mathematically you can say the volume went from 1 to 8 but the surface area only went from 6 to 24. As volume increased 8 times, surface area only increased 4 times.

Now apply this to living systems. For example, heat exchange. In a living organism you can say that heat is exchanged between the organism and its environment through it's surface. For a human you could say heat is exchanged through the skin. So, who is going to lose heat faster, a baby or an adult? A baby is obviously smaller volume, but it has relatively higher surface area. As the baby grows into an adult it's volume increases at a faster rate than the surface area. Therefore, an adult has more volume to produce heat, and relatively less surface area to lose it through. In other words, the baby loses heat much faster because it has more RELATIVE surface area!

Conclusion:

Area = side x side

Surface area = Area of all sides added up

Volume = side x side x side

As volume increases, surface area increases less

Apply it. If you took one breath and filled up one big balloon, or used the same breath to fill 10 small balloons, how would the volume compare? How would the surface area compare?