Math Problem of the Week: Finding the Surface Area to Volume Ratio of a Sphere

One of the tenets of cellular biology is that as the volume of a cell increases, its surface area to volume ratio decreases. After a certain volume has been attained, this spells trouble for a cell, because the surface area is too small to allow for adequate diffusion of nutrients and wastes per unit of time. How can we demonstrate this mathematically?

For starters, we note the following formulas:

Surface area of sphere = 4πr^2

Volume of sphere = (4/3)πr^3

This being the case, we note that the Surface Area to Volume ratio is:

SA / Vol = (4πr^2) / [(4/3)πr^3]

Therefore, the factors π and 4 in numerator and denominator cancel, as well as r^2 in both numerator and denominator, to give:

SA / Vol = 3 / r

Thus, the surface area to volume ratio is clearly inversely proportional to r. Therefore, a larger volume means there is less surface area in relation to the volume. No wonder why there are volume limitations in cells; beyond a certain size threshold, there is simply not enough surface area for nutrients and waste to diffuse through relative to the volume!