Concepts of Biology (BIOL115)  Dr. S.G. Saupe (ssaupe@csbsju.edu); Biology Department, College of St. Benedict/St. John's University, Collegeville, MN 56321 
SURFACETOVOLUME RATIOS IN BIOLOGY
These exercises are designed to introduce you to the concept of surfacetovolume ratios (S/V) and their importance in biology. S/V ratio refers to the amount of surface a structure has relative to its size. Or stated in a more gruesome manner, the amount of "skin" compared to the amount of "guts." To calculate the S/V ratio, simply divide the surface area by the volume. In this exercise we will first examine the effect of size, shape, flattening an object, elongating an object on surfacetovolume ratios. Then, we will do some handson activities to examine the importance of S/V ratios in two areas, cell size and metabolic rate. Equations to help with the calculations are provided at the end of the exercise.
You will be asked to complete an assignment for this activity:
PreClass Assignment:

PostClass
Assignment:

EXERCISE 1. INFLUENCE OF SIZE ON S/V RATIOS.
We will use a cube to serve as a
model cell (or organism). Cubes are especially nice because surface area (length x width x
6 sides) and volume (length x width x height) calculations are easy to perform. To
calculate the surfacetovolume ratio divide the surface area by the volume. Complete the
table below for a series of cubes of varying size (equations):
Table 1. Effect of increasing size on surfacetovolume ratio  
Length of a side (mm) 
Surface Area (mm^{2}) 
Volume (mm^{3}) 
Surface/volume ratio 
1 

2 

3 

4 

5 

6 

7 

8 

9 

10 
Questions and Analysis:
EXERCISE 2. SHAPE AND S/V RATIOS:
In this exercise we will explore the
impact of shape on surface to volume ratios. The three shapes given below have
approximately the same volume (1 mm^{3}). Complete the calculations for each shape.
To calculate the volume of environment within 1.0 mm of the shape, imagine the
shape surrounded by a larger shape of the same size that is 1.0 mm bigger on all
sides. (equations)
Table 2. Effect of shape on surfacetovolume ratios  
Shape  Dimensions (mm)  Volume (mm^{3})  Surface Area (mm^{2})  S/V ratio  Volume of environment within 1.0 mm 
Sphere  1.2 diameter  
Cube  1 x 1 x1  
Filament  0.1 x 0.1 x 100 
Questions & Analysis:
EXERCISE 3. S/V RATIOS IN FLATTENED OBJECTS:
In this exercise we will explore
how flattening an object impacts the surface to volume ratio. Consider a box that is 8 x 8
x 8 mm on a side. Then, imagine that we can flatten the box making it thinner and thinner
while maintaining the original volume. What will happen to the surface area and s/v ratio
as the box is flattened? Complete the table below. (equations)
Table 3. Effect of flattening an object on surfacetovolume ratio  
Box No. 
Height (mm)  Length (mm)  Width (mm)  Surface area (mm^{2})  volume (mm^{3})  s/v ratio 
1 
8 
8 
8 

2 
4 
16 
8 

3 
2 
16 
16 

4 
1 
32 
16 

5 
0.5 
32 
32 
Questions/Analysis:
EXERCISE 4. S/V RATIOS IN ELONGATED OBJECTS:
In this exercise we will explore
how elongating an object impacts the surface to volume ratio. Consider a box that is 8 x 8
x 8 mm on a side. Then, imagine that we pull on the ends to make it longer and longer
while maintaining the original volume. What will happen to the surface area and s/v ratio
as the box is flattened? Complete the table below. (equations)
Table 4. Effect of elongating an object on surfacetovolume ratio  
Box No. 
Length (mm)  Height (mm)  Width (mm)  Surface area (mm^{2})  volume (mm^{3})  s/v ratio 
1 
8 
8 
8 

2 
16 
4 
8 

3 
32 
4 
4 

4 
64 
2 
4 

5 
128 
2 
2 
Questions/Analysis:
EXERCISE 5. WHY ARE CELLS SMALL?
The typical eukaryotic cell is rather small 
approximately 100 mm in diameter. This exercise is designed to help provide an explanation
why cells are not normally larger.
Obtain 2 cell models, one small and one large. Measure the length and diameter of each and then record your data in the table below. Place each cell in a bowl containing clear vinegar (BE CAREFUL!). Allow to sit for a few minutes, or until most of the blue color is gone from the smallest cell. Remove the models with a plastic spoon (CAUTION: don't get the vinegar on your hands!!!!) and place it on a piece of paper towel. Then, measure the size of the colored area remaining and record these data in the table below. Complete the calculations.
Theory: The cell models are made of a gelatinlike material called agar. The agar has an acidsensitive dye incorporated into it. The dye turns from blue to yellow (clear in the presence of acid). The uptake of acid, and hence colorless areas of the cell models represents the uptake of food/nutrients by the cell. From this, we can calculate the percent of each cell that was fed during the incubation period.
Table 5. Effect of cell size on feeding rates (equations)  
Small Cell 
Large Cell 

Colored Portion Before Feeding (initial)  diameter (mm)  
radius (mm)  
height (mm)  
surface area (mm^{2})  
volume (mm^{3})  
S/V ratio  
Colored Portion After Feeding (final)  diameter (mm)  
radius (mm)  
height (mm)  
surface area (mm^{2})  
volume (mm^{3})  
Percent of volume (cell) fed = (initial volume  final volume)/initial volume x 100 
Analysis/Questions:
EXERCISE 6. WHY DO MICE HAVE GREATER METABOLIC RATES THAN ELEPHANTS?
It is
well known that there is an inverse relationship between body size and metabolic rate. The
purpose of this exercise is to determine the reason for this relationship.
Procedure:
Heat content (joules) = temp. (C) x vol. (cm^{3}) x specific heat of water (4.2 joules/cm^{3} C)
Calculate the total heat loss during the two minute period (row 11) by subtracting the
final heat content (row 10) from the initial heat content (row 8).
Table 6. Effect of size on the rate of cooling (equations)  
Small 
Large 

1. height (cm)  
2. diameter (cm)  
3. radius (cm)  
4. volume (cm^{3})  8 
80 
5. surface area (cm^{2})  
6. S/V ratio  
7. initial temp (C)  40 
40 
8. initial heat content (J)  
9. final temp (C)  
10. final heat content (J)  
11. total heat loss (Joules/2 minute)  
12. total heat loss (Joules/min)  
13. relative heat loss (J/min/cm^{3}) 
Analysis/Questions:
ADDITIONAL QUESTIONS:
(answer one of these for your post assignment)
References:
Useful Equations:
Shape  Equation for Volume  Equation for Surface Area 
cube  l x w x h  l x w x 6 
box (filament)  l x w x h  determine SA for each side then add 
sphere  4/3 (p) r^{3} = 4.189 r^{3}  4 (p) r^{2} = 12.57 r^{2} 
cylinder  p r^{2 }l  2 p r^{2} + 2 p r h = 2 p r (r + h) 
Supplies Required:
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