Surface Area to Volume Ratio Tutorial
In order to understand why smaller cells are more efficient than larger cells, we must examine the
mathematical relationship between the surface area of a cell (plasma membrane) and its volume (the
cytoplasm.) This is referred to as the surface area to volume ratio, or SA:V, for short.
Lets examine a hypothetical cube-shaped cell.
I have chosen a cube because (1) it is a reasonable approximation of the shape of some cells and (2)
it’s geometrical dimensions are easier to calculate.
What is the formula for the surface area of a cube: L x W x 6
The area of one side of a cube is simply its length (L) multiplied by its width (W). A cube has six
sides. All of the surfaces (sides) of a cube are the same size so we can multiply the area of one side
by six to get the surface area.
What is the formula for the volume of a cube: L x W X H
The volume of any box is the length (L) multiplied by its width (W) multiplied by its height (H).
Let’s say that the cube above has the following measurements:
L = 1 cm
W = 1 cm
H = 1 cm
With these measurements we can obtain the surface area and the volume:
Surface Area = L x W x 6 = 1 cm x 1 cm x 6 = 6 cm2
Volume = L x W x H = 1 cm x 1 cm x 1 cm = 1 cm3
In order to understand how these measurements impact cells, we are not concerned as much with
the actual values of surface area and volume as much as we are concerned with their ratio. A ratio
is a proportion, a comparison, of the size of one thing to another. In this case, we want to see
proportionally how much larger the surface area of the cell is compared to its volume.
To write these values as a ratio, we simply put the values together like this:
Surface Area : Volume
SA : V
Lets use the values we calculated for our hypothetical cell to find its surface area to volume ratio.
SA : V
6 cm3 : 1 cm2
6: 1
Because this is simply a comparison and not a measurement, we can drop the units and simply write
the ratio as 6 : 1. This means that for comparison purposes, the magnitude of the surface area is
six times larger than the volume.
Great! So what does it mean? Well … nothing, unless you compare it to the surface area to volume
ratios of other cells.
Calculate the surface area to volume ratios (SA : V) a cells that has a length, width, and height of 2
cm.
Surface Area = L x W x 6 = 2 cm x 2 cm x 6 = 24 cm2
Volume = L x W x H = 2 cm x 2 cm x 2 cm = 8 cm3
Surface Area to Volume ratio (SA: V) = 24 : 8
Is 24 : 8 a bigger or smaller ratio than that of our first cell with a ratio of 6 : 1? To figure this
out, it might be helpful to simply the ratio. Dividing both numbers in the ratio by the volume will
make the volume number “1.” This will make it easier to compare. For instance:
24 : 8
8
8
= 3 : 1
You can see that 6 : 1 is a larger ratio than 3 : 1. What does this mean? It means that the second
cell has a smaller surface area compared to its volume than the first cell. If you continue to make
the cell larger, you would notice that the ratios continue to decrease. If you remember that the
surface area, the plasma membrane, of the cell is critical to bringing in nutrients and expelling
wastes from the cell, then you realize that a smaller ratio is more efficient for the cell.
In other words, a smaller cell is more efficient than a larger cell!
Surface Area to Volume Ratio Practice
1.
Examine the cubes provided. There are six colors (pink, orange, yellow, green, blue and gray).
2. Make a hypothesis about which cube will have the highest surface area to volume ratio. Which
will have the smallest SA:V ratio?
3. Rank each colored cube in order of greatest surface area to volume ratio down to smallest
surface area to volume ratio in the Cube Color column of the chart below.
4. Each lab group should select ONE of the cubes and using a ruler, determine its surface area and
volume.
SURFACE AREA:
VOLUME:
5. Calculate your cube's surface area to volume ratio (SA:V).
6. Place the information you calculated in #4 and #5 into the chart below.
7. Obtain the information for the remaining colored cubes from other groups. Complete your
chart.
8. Were the rankings you made in #3 correct? Why or why not?
Cube Color
Surface Area
Volume
SA:V
Ratio Size
Largest
Smallest