Surface Area vs Volume

So if both of these quantities increase as cells or balloons enlarge, why does it matter? Well, let's think about the units. surface area is measured in meters squared, whereas volume is measured in meters cubed. So, let's ignore the balloon for now and think about a simpler shape - a cube. In the image shown, the cube on the right is just like any cube with an equal length of all sides.

In this particular case, that length is 1 cm (the units are not listed in the image because it is best to focus on just the number for now). If the side length is 1 cm, then the surface area of that cube on the right is 6 centimeters squared, because there are six sides to a cube. You have to find the area of one side (1 cm x 1 cm = 1 cm2), and multiply it by those 6 sides. So the total surface area of the right cube is 6 cm2. The volume is calculated by multiplying length x width x height. This is, of course, 1 cm x 1 cm x 1 cm = 1 cm3.

Now, let's look at the cube on the left. The length of this cube is 3 cm. So the sides are 3 times as long as those of the first cube. As a result, we can calculate the surface area and volume just like before. Doing so yields a volume of 27 cm3 and a surface area of 54 cm2 .

If you look closely, you can see that the shape's volume is increasing at a faster rate than its surface area. That's because surface area is calculated by squaring a side length (and multiplying by 6), whereas volume is calculated by cubing a side length. So doubling a side length actually increases surface area by a factor of 4 (because double is 2, and 2 squared is 4), whereas it increases the volume by a factor of 8 (because 2 cubed is 8). So volume increases much faster than surface area.

This is referred to as the surface law, and it explains a shocking amount about you. Think about a balloon: when a balloon is empty, it is mostly rubber with little to no air inside. When it is blown up, it becomes mostly air with a relatively small amount of rubber on the outside. "The more you inflate it, the more its interior dominates the whole."  One implication of the importance of the surface law is with body temperature. Heat is lost at the surface, so the more surface area you have relative to volume, the harder it is to stay warm. Source: The Body: A Guide for Occupants by the excellent Bill Bryson.

Surface Area/Volume Ratios

Okay, now we can see that volume increases faster than surface area does. Why do I care? Well, a very important number when it comes to cells (and everything from food to roots) is their surface-area-to-volume ratio. Have you ever noticed how you taste food a lot more after you've chewed on it a bit? You are breaking a large cube (let's simplify to cubes again) into a lot of smaller cubes. That means a lot more surface area is able to interact with your tongue. This means you taste more when there is more surface area.

This ratio matters for cells because the surface area is the way by which a cell interacts with its environment - this is how it obtains its resources. If a cell needs to take in some molecule, that molecule can only get in by interacting with the surface area (the cell membrane). If a cell is large, it needs more resources. So, it needs a higher surface area. If it gets so big that the volume is horrendously large, the resources needed by the cell are equally horrendous. As it grew, the cell was able to take in more resources due to its higher surface area. But the demand increases faster than the supply, because volume increases faster than surface area.

If you need to calculate the surface-area-to-volume ratio in order to compare two cells, you simply need to divide surface area by volume.