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Thinking about the Surface Area To
Volume Ratio
Why are living organisms the
sizes that they are?
Are there advantages to being large or small? What are the advantages
to being a certain size?
When you make the outside of something
bigger, interesting things happen to the inside. A little simple
math can help us to see what happens. Then we will go on to dinosaurs
and mice!
Let's look at some math examples.
Area
Here is a cube. Each edge measures two inches long.
How do we find the area of the surface, the outside?
We multiply two touching edges together.
Each edge is 2 inches long, so the area of one flat side of
the cube is
2*2 = 4 square inches.
The area of one side of the cube is 4 square inches.
A cube has 6 sides.
4 square inches * 6 sides = 24 square inches.
So now we know the surface area of the outside of the whole
cube.
Volume
Now let's think about the inside.
To find out the volume (how much is inside) of the cube we
multiply height by width by length.
With this cube, we multiply 2*2*2 = 8 cubic inches.
So we have 24 square inches of surface to only 8 cubic inches
of volume.
Is the surface always a larger number? Let's do some more
cubes and see.
| side of cube |
side*side |
Area of Side |
side*side*6 |
Area of Cube's Surface |
side*side*side |
Volume |
Ratio of Surface Area to Volume |
|
2
|
2*2
|
4
|
2*2*6
|
24
|
2*2*2
|
8
|
3 to 1 |
|
4
|
4*4
|
16
|
4*4*6
|
96
|
4*4*4
|
64
|
3 to 2 |
|
6
|
6*6
|
36
|
6*6*6
|
216
|
6*6*6
|
216
|
3 to 3 |
|
8
|
8*8
|
64
|
8*8*6
|
384
|
8*8*8
|
512
|
3 to 4 |
|
12
|
12*12
|
144
|
12*12*6
|
864
|
12*12*12
|
1728
|
3 to 6 |
|
20
|
20*20
|
400
|
20*20*6
|
2400
|
20*20*20
|
8000
|
3 to 10 |
Let's look at the numbers here. I
notice something very odd. When the inside of the cube (the volume)
is small, the outside (the surface area) is large by comparison.
What does this mean?
In practical terms for world builders,
it means that small animals, like mice, have a lot of skin on
the outside compared to a small volume of bones, blood, and internal
organs on the inside. So a mouse or a baby bird looses heat rapidly.
When an animal has a large interior
volume (big bones, muscles, lungs, digestive system) the skin
on the outside (the surface area) gets progressively smaller
when it is compared to the volume inside. This means that large
animals tend to maintain their temperatures and to get cooler
or warmer slowly. It is also difficult for them to get rid of
excess body heat, for example, if they have been running.
Let's Apply These Ideas
Single-Celled Organisms
Why are single-celled organisms so
small? Look at the table above. Tiny organisms have to take in
food, water, gases, and chemicals, and they have to excrete waste
products. They are full of organic molecules that have to be
moved around and processed. In small cells, all parts of the
cell are near the cell membrane, and chemical exchanges with
the environment are simplified. As single cells grow larger,
the contents increase in volume, and making everything work together
becomes more complex. The cell wall, the interface between the
living cell and the world, does not increase in area as quickly
as the volume of the cell contents. At some point, the structures
and processes in the cell will become inefficient if the cell
becomes too large.
A sphere is a strong, efficient shape.
Cells that are shaped like tiny bubbles contain the greatest
volume for their surface area. However, cells come in a great
variety of shapes. We have seen many cells that are shaped like
tiny rods. Rod-shaped cells will have more surface area compared
to their contents than spherical ones. However, as cells evolve
to increase their surface area, they risk becoming increasingly
fragile.
Small Animals
If you look at the table above, you
can see that very small organisms have a lot of surface compared
to their volumes. Small warm-blooded animals (endotherms), like
mice and hummingbirds, have to eat a lot in proportion to their
weights in order to keep themselves warm. In cold weather, a
small animal loses heat rapidly because so much of it is in contact
with the environment. It needs to eat a lot, stay in a warm area,
and have good insulating fur or feathers.
On earth many small animals are cold-blooded
(exothermic). Amphibians and fish, reptiles and insects, all
deal with being small by metabolizing at the temperature of the
environment. A small cold-blooded animal can warm up quickly
on a sunlit rock, and requires little food to stay alive. This
is certainly efficient. However, cold-blooded animals react more
slowly than warm-blooded ones when they are cold. They need to
find refuges safe from predators as the environment cools.
Large Animals
Being large also has consequences.
Large animals can hold heat well, but may have trouble getting
rid of body heat in very hot weather. This is an important problem,
because proteins break down if the body gets too hot, and the
organism may die if overheated. Many animals have developed ways
of getting rid of extra heat by sweating or panting. Elephants
can dissipate heat from their large, thin ears.
There is some disagreement about whether
the dinosaurs were warm blooded or cold blooded, or in some intermediate
stage. A large, warm dinosaur body could probably stay warm overnight,
but if it ever got really chilled, it would take a very long
time to warm it up!
Being large poses problems for supporting
structures in the body. If you look at the table above you can
see that if you double the measurements and then multiply the
size by itself (this is called squaring the side, e.g.,
6*6) you certainly get more surface area. However, when you multiply
to get the volume you multiply side one by side two by
side three, e.g., 6*6*6, which is called cubing the number. Therefore
you get an important increase in volume, and weight. (We have
used cubes for our example, but irregular shapes follow the same
pattern of increase.) A larger animal needs more massive bones.
More body cells need to be nourished: that means that the animal
needs to have, and pump, more blood. It will need to enlarge
lung size, and increase digestive capacity to get the extra food
to support a larger body.
Animals adapt to their environments.
Ponies are horses who have developed smaller bodies so that they
can live where the pasture is sparse. The Science Series Nova has a program on the extinction
of the mammoth. The last mammoths were pygmies, only three feet
tall, living in an environment where there was not enough food
to support their larger ancestors.
Things to Think About
A small increase in size leads to
a larger increase in weight. Think about this when you are designing
flying creatures.
What effect does the gravity of your
planet have on the weight of your life forms? How will that affect
their structure?
Remember that an environment must
provide enough food to support its animals.
Where would you expect
to find large animals?
Where would you expect only small ones?
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