Greenwood press

inefficient way of cutting
cutting circles with less 
pattern for cutting a bandage
circles (18 circles)
waste (22 circles)
box
Cutting shapes from sheets of material that will be used in packaging.

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wax (surface area) for the necessary storage area (volume of the cell). The sur-
face area of the cell is given by 
S = 6sh −
3
2
s
2
cot(θ) +

3s
2
√
3
2
 csc(θ), where
S is the surface area, s is the length of the sides of the hexagon, and h is the
height. The values of 
s and h are constant for specific species of bees. Using cal-
culus, the angle that requires the least volume of wax for cells has a size of 55°
regardless of 
s and h. Measurements of the actual angles in hives rarely differ
from this value by more than 2°.
Nature sometimes needs to maximize surface area. The interiors of your
lungs are networked with air sacs (alveoli). The sacs are formed from very thin
membranes that allow oxygen to pass from the air in the lungs to your blood, and
carbon dioxide to move from your blood to the air that will be exhaled. The sur-
face area covered by a human’s skin is about 2 square meters, but the total sur-
face area of the alveoli is about 100 square meters! The massive surface area is
needed to provide sufficient exchange of the two gasses within the time of one
breath. Similarly, fish have gills that offer substantial surface membranes
between the water and the bloodstream so that they can quickly exchange the car-
bon dioxide in blood for oxygen from the water. 
Some common household tasks favor larger areas. If you want to dry wet
clothes, you should spread them out rather than rolling them into a ball. If you
want to cool a drink fast, crush an ice cube into the beverage rather than drop-
ping a solid cube into it.
The fundamental law of similarity asserts that when you scale up (or down)
a solid figure by a scale factor 
k, you scale up the surface area by k
2
and the vol-
ume by 
k
3
. If you build a car model that is a 1:24 scale model of a real car, that
means you are multiplying each dimension of the car by 1/24. The surface area
would be changed by a factor of (1/24)
2
, and the volume by (1/24)
3
. If the model
and the real car were made from the same materials, then the weight scale would
match volume. Weight would be scaled down by (1/24)
3
. (See Ratio.) Because
scaling has such a dramatic influence on surface area and volume, larger animals
have an easier time maintaining their metabolism levels than do smaller ones.
This can be shown by examining the ratio of volume to surface area for a series
of cubes, starting with 1 cm on a side through 1 meter on a side.