Greenwood press

side (cm)
area (cm
2
)
volume (cm
3
)
ratio of volume to area
1
6
1
1/6 ≈ 0.17
3
54
27
27/54 ≈ 0.5
10
600
1,000
1,000/600 ≈ 1.67
50
15,000
125,000
125,000/15,000 ≈ 8.33
80
38,400
512,000
512,000/38,400 ≈ 13.33
100
60,000
1,000,000
1,000,000/60,000 ≈ 16.67
Ratio of volume to area for different cubes.
134
SURFACE AREA

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An animal loses heat through external surface area. The energy needed for
basic metabolism is roughly proportional to the volume of the animal. An animal
that has a large volume with respect to its surface area will have an easier time
maintaining its metabolism. A large animal will have a lower heart rate and food
requirements that are a fraction of its weight. An animal that has a small volume
compared with surface area will have to work hard to replace the heat lost. We
can expect small animals to have rapid heartbeats and daily food requirements
that may be multiples of the animals’ weight. A polar bear is compact: Its large
bulk means that it will have a large volume-to-area ratio. A hummingbird has a
volume-to-surface-area ratio close to 0.5. It loses heat rapidly and hence must
have a rapid heartbeat and relatively large food intake to maintain its metabolism.
Perhaps that is one of the reasons that there are no hummingbirds in the Arctic. 
Of course, animals aren’t cubes. A human being is not packaged like a polar
bear. We have hands that are very useful, but in winter they provide more surface
area proportional to the volume they contain, so the wise person will wear mit-
tens instead of gloves to reduce the heat loss. 
The volume-to-surface-area ratio is a factor at the microscopic level. Cells
cannot benefit from larger volume-to-surface-area ratios. Since energy must come
through the cell membrane, small ratios are an advantage. The compensation
made by plant cells is that larger plant cells maintain less of a spherical shape
(more cylindrical), while small plant cells are close to spherical. With a less spher-
ical shape, the larger cells maintain an advantageous volume-to-surface-area ratio.
online sources for further exploration
Camping (wearing gloves)

Figuring out how many rolls of wall covering you need

Lead paint concentrations

Medicinal tablet surface area

Minimum surface area of a can

Painting

Soap bubbles


Size effects on airplane lift

Theory of flight