Surface area to volume ratio is an important factor we need to consider if we want to understand the movement of biological molecules in and out of organisms.
An object's surface area is the area of its outer surface. It is measured in units of distance squared - such as cm2 or m2.
An object's volume is the amount of space that the object occupies. It is measured in units of distance cubed - such as cm3 or m3
An object's surface area to volume ratio is the ratio of its surface area to its volume. We can represent ratios by writing a colon (:) between the two things that we are taking the ratio of. So surface area to volume ratio can be represented by writing surface area : volume.
As an example, let's say that an object has a surface area of 2m2 and a volume of 10m3. The object's surface area to volume ratio is therefore:
2 : 10
However, it is conventional to write the ratio with the last number as 1. In other words, to show how much surface area there is per unit of volume.
To get from 10 to 1 we need to divide by 10. Therefore, we divide both numbers by 10 to get an equivalent ratio to the one above but with 1 as the second number:
0.2 : 1
This tells us that the object has 0.2m2 of surface area for every 1m3 of volume.
Consider a cube with sides that are 1cm long.
The area of each face of the cube is 1cm x 1cm = 1cm2. There are 6 faces, so the total surface area is 1cm2 x 6 = 6cm2.
The volume of the cube is 1cm x 1cm x 1cm = 1cm3.
Therefore, the surface area to volume ratio of the cube is 6 : 1.
Now, think about what happens if the size of the cube is increased so that its sides are 10cm long.
The surface area is now 10cm x 10cm x 6 = 600cm2.
The volume is now 10cm x 10cm x 10cm = 1000cm3.
Therefore, the surface area to volume ratio is now 600 : 1000. If we divide both sides by 1000 we find that this is equivalent to 0.6 : 1.
Finally, consider what happens if we increase the side length to 100cm.
Now the surface area is 100cm x 100cm x 6 = 60,000cm2.
The volume is 100cm x 100cm x 100cm = 1,000,000cm3.
The surface area to volume ratio is 60,000 : 1,000,000. If we divide both sides by 1,000,000 we find that this is equivalent to 0.06 : 1.
The surface areas, volumes and surface area to volume ratios of the three sizes of cube are summarised in the table below:
|Cube side length (cm)||Surface area (cm2)||Volume (cm3)||Surface area : volume|
|1||6||1||6 : 1|
|10||600||1000||0.6 : 1|
|100||60,000||1,000,000||0.06 : 1|
We can see that as the size of the cube increases, its surface area to volume ratio decreases. This is true for any object.
This is because although both the surface area and the volume increase as the object gets bigger, the volume increased by a bigger scale factor than the surface area does for a given increase in the size of the object.
For example, every time the cube's sides get 10 times longer, the surface area becomes 100 times greater, but the volume becomes 1000 times greater.
This is because as the side length increases, the volume is stretched out in three dimensions, whereas the surface area is only stretched out in two dimensions.
Mathematically speaking, the surface area of an object is proportional to the length squared, whereas the volume is proportional to the length cubed, meaning that the volume increases at a greater rate than the surface area does as the length increases. This means that as the object gets bigger its surface area to volume ratio decreases.
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What happens to an object's surface area to volume ratio as its size increases?
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