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An Analysis of the Surface to Volume Relationship

Buzz Hoagland
Biology Department
Westfield State College
Westfield MA 01086
bhoagland@westfield.ma.edu

Casual observations lead us to believe that large objects cool more slowly than do small objects of similar shape. For example, if you place two rocks of similar shape, but different masses, in the sun for a few hours and then move them to the shade, the smaller of the two will cool off more quickly. This phenomenon is believed to be a result of the time it takes the internal heat of each rock to dissipate to the cooler environment. Because the distance from the center of the larger rock to the surface is greater than that of the smaller rock, it should cool off more slowly. Expressed another way, the larger rock has a much smaller surface area relative to its volume than does the smaller rock.

For similarly shaped geometric objects, surface area increases with the square of the linear dimensions and volume increases with the cube of the linear dimensions. Therefore, we expect for an array of similarly shaped objects that volume will increase faster than surface area. In fact, based upon simple geometric principles, we can predict the precise relationship between surface area and volume: surface area is related to volume raised to the power 2/3, or 0.67.

The purpose of this investigation is to demonstrate graphically that larger objects of similar shape have less surface area relative to their volume as compared with smaller objects.



Materials and Methods


Microsoft Excel was used to generate 100 uniformly distributed random numbers between 1 and 100.The numbers were squared and cubed to represent surface area and volume, respectively. Surface area and volume values were then plotted with surface area representing the independent variable and volume representing the dependent variable.



Results


It is readily observed that volume increases at a faster rate than does surface area for similarly shaped objects (Figure 1), and that the relationship between these two variables is surface area = volume0.67.

Figure 1. The relationship between surface area and volume. See text for an explanation of how this figure was generated.


Discussion


The surface-to-volume ratio plays an important role in topics as diverse as designing canoes to the treatment and care of hypothermia. Physiologists have investigated the theoretical relationship of surface-to-volume ratios and metabolism for much of this past century. They have documented the relationship between resting metabolic rate and body mass in hundreds of species from unicellular organisms to homeotherms (Schmidt-Nielsen, 1997). Despite the very low metabolic rates in unicellular organisms and poikilotherms, the relationships between metabolic rate and mass are consistent. The slopes of the regression lines between metabolic rate and body mass are approximately 0.75. Presumably, resting metabolic rate is directly proportional to rate of heat loss, and rate of heat loss should be directly proportional to surface to volume ratios. That being the case, we should expect the slopes of the regression lines to be 0.67 as demonstrated above. However, because 0.67 is significantly different from 0.75 (Schmidt-Nielsen, 1997), it can be concluded that temperature regulation is not the primary cause of the observed relationships between metabolic rate and body mass.

While it is true that rocks obey simple surface to volume realtionships when it comes to heat transfer, it is equally true that living organisms are much more complicated. Although, large people will not experience cold the same way small people do, this relationship is not as simple as the physics of surface to volume relationships. Nutritional status, body position, cold tolerance, and many other factors must also be accounted for in this relationship.



Literature Cited



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