If two objects have the same shape, they are said to be geometrically similar. By definition, the ratio of any two linear dimensions of one object will be same for any geometrically similar object. This is easiest to illustrate with simple geometric shapes:
For each pair of geometrically similar shapes above, the proportions are identical. That is, the ratio of width to height of the rectangles is 2.0, and the ratio of the height to the hypotenuse of the triangles is 0.6. Notice that all of the dimensions of one can be calculated by multiplying the dimensions of the other by a constant, in these examples 2 (or 1/2). If you make a geometrically similar rectangle three times as high as the small one above, then its width will be three times as big as well, along with its diagonal measurement and perimeter, i.e. ANY linear dimension will be multiplied by the same factor.
What about other characteristics, such as surface area and volume? Do they show the same relationships as linear measurements? If you double the linear dimensions of an object, do you double its surface area and volume also? Would the ratios of volume to surface area or volume to length remain constant? To answer these questions, imagine that you could build the three cubes illustrated below using sugar cubes 1 cm on a side. The first figure is a single sugar cube, the second consists of 8 sugar cubes, and the third is assembled from 27 sugar cubes.
Fill in the table below (on your answer sheet) with calculations for each of the three cubes:
small cube medium cube large cube length (cm) surface area (cm2) volume (cm3) surface area/volume surface area/length volume/length
You can determine the areas and volumes intuitively by counting the exposed faces and imagining the hidden faces. Alternatively, you could use the basic formulas for surface area and volume of a cube:
surface area = 6L2
volume =L3
By now you should be aware that for these cubes, at least, the surface area to volume ratio decreases as the cube gets larger. Not exactly rocket science, and what might this have to do with organisms? More than you might imagine. Nothing is more basic to life than the exchange of materials, and in virtually all cases in biology this exchange occurs through surfaces. Blood exchanges oxygen and carbon dioxide with the atmosphere through the surface of the lung (the alveoli) and with the tissues of the body through the walls of the capillaries. Nutrients enter your body through the surfaces of the stomach and small intestine; wastes leave the blood through the surfaces of tubules in the kidneys. Anything that enters or leaves a cell must pass through the cell membrane, another two dimensional surface. Such processes will obviously be limited, at least in part, by the available surface area through which exchange may occur. On the other hand, the rate at which all of these process of exchange must occur is dictated by the volume of the cell or tissues that are served - total demand depends on the biomass that must be supported. The quantitative relationship between surface area and volume is thus of primary biological importance.
The relationships between lengths, areas, and volumes of geometrically similar objects can be predicted rather simply. Areas of an object (such as the surface area or cross sectional area) are proportional to the square of the object's length (L2), while its volume is proportional to the cube of its length (L3). Thus, ratios of lengths, areas, and volumes will be proportional to length raised to some power:
Since Surface Area 
L2
and Volume L3,
SA/V
For the cubes shown above, the SA/V of each cube is 6 /L. The "6" in this case is a constant that works for cubes; other constants would be used for different shapes, but the 1/L works for any shape).
Simply restated, surface to volume ratios are inversely proportional to size (or length). This geometrical observation has profound biological implications; as Julian Huxley once observed, comparative anatomy is primarily the story of the evolutionary struggle to maintain surface to volume relations. We will explore some aspects of this story in lecture.
Practice Questions (click letter to test your response)
1. How would you predict the ratio of Surface Area to Length to vary with size?
a. Surface Area/Length
b. Surface Area /Length
c. Surface Area /Length
d. Surface Area /Length
2. How would you predict the ratio of Volume to Length to vary with size?
a. Volume/Length
b. Volume/Length
c. Volume/Length
d. Volume/Length
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