Greenwood press

132
STEP FUNCTIONS

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SURFACE AREA
133
SURFACE AREA
There are more uses of surface area than determining how much paint to buy
to paint a house. The mathematics of surface area determines how objects retain
heat, how cans are cut from sheets of metal, how cells exchange fluids, and how
animal metabolism relates to size. Two important mathematics questions about
surface area are: “What shapes make surface area a minimum for a specific vol-
ume?” and “For the same shape, how do volume and surface area change as the
figure is scaled up or down?”
The first question has some simple results for common figures. The cube is
the solid that minimizes surface area for a specific volume in a prism. The sphere
is the solid that minimizes surface area for any volume. This last result shows up
in soap bubbles or oil drops. In the absence of other forces, these will be spheres.
Packaging companies have additional minimization issues to handle when
they determine how a package such as a cereal box or a soda can should be con-
structed from raw materials. The desired volume is not the only issue they must
consider. If the product is going to grocery stores, then it has to have standard
dimensions. The shape of the product may determine or restrict the dimensions
of the package. If the carton is glued together, then additional surface is needed
for the glued regions. Finally, most packaging is cut from one piece of flat mate-
rial, so the engineer has to decide how the cuts will be made to minimize waste.
Some of the issues have natural solutions. For example, the first illustration in
the figure below shows a wasteful method of cutting circular-can lids from sheets
of aluminum. The middle diagram shows that stacking the circles like the cells
in a beehive would produce four more lids from the same sheet of material. The
complexity of cutting single cartons is shown by a flattened box of bandages in
the last illustration. Many of these cartons must be cut from large pieces of
glazed cardboard.
Nature has solved the minimization issue in remarkable ways. In a beehive,
each cell is a regular hexagonal prism, open on one end and with a trihedral angle
at the other. The trihedral angle must have a consistent geometry, because the
bees build identical cells on the other side of one wall of cells. It is believed that
this shape developed because it is strong and because it uses the least amount of