Surface Area to Volume Ratios (Cambridge (CIE) A Level Biology): Revision Note

Calculate the surface area-to-volume ratios of the two following microorganisms:

1. A bacterial cell from the species Staphylococcus aureus; you can assume that each cell is a cube with side length of 800 nm (8 × 10^-7 m)

2. A bacterial cell from the species Bacillus subtilis; these are rod-shaped cells which you can assume to be cylindrical in shape. They are 5 µm long and 1 µm in diameter

Comment on your calculated answers. 

Solution

1. For the Staphylococcus aureus cell: side length = 800 nm

Convert this value into µm by dividing by 1000

$\frac{800\mathrm{nm}}{1000}=0.8\mathrm{\mu m}$

Surface area of cube = 6(s²)

= 6 × (0.8 × 0.8) = 3.84 µm²

Volume =  0.8  × 0.8 × 0.8

= 0.512 µm³

The SA:V ratio is the surface area divided by the volume, so

$\mathrm{SA}:\mathrm{V}\mathrm{ratio}=\frac{3.84}{0.512}=\frac{7.5}{1}=7.5:1\left(\mathrm{or}\mathrm{simply}7.5\right)$

___________

2. For the Bacillus subtilis cell: radius = 0.5 µm (half of diameter 1µm) 
Surface area of a cylinder =  2πrh + 2πr²
= (2π × 0.5 × 5) + (2π × 0.5²)
= 15.708 + 1.571 µm²
= 17.278 µm²

Volume of a cylinder: formula is πr²h
= π × 0.5² × 5
= 3.927 µm³

$\mathrm{SA}:\mathrm{V}\mathrm{ratio}=\frac{17.278}{3.927}=\frac{4.4}{1}=4.4:1\left(\mathrm{or}\mathrm{simply}4.4\right)$

The cuboid cell of Staphylococcus aureus has a surface area-to-volume ratio of 7.5 : 1

The cylindrical cell of Bacillus subtilis has a surface area-to-volume ratio of 4.4 : 1

The smaller cell has the largest surface area-to-volume ratio of the two shapes. This allows efficient diffusion into / out of the cell. The larger the surface area to volume ratio, the more membrane there is per unit volume, so exchange by diffusion can keep up with the needs of the whole cell. So the larger the surface area to volume ratio, the better adapted to simple diffusion that organism is. 

This worked example assumes that Staphylococcus aureus is a cuboid shaped cell; in fact it is a spherical cell.

You are not required to know how to calculate the surface area and volume of a sphere for this exam. For a given volume, a sphere has the smallest surface area of any shape, so cells that rely on diffusion stay small, or are flattened or elongated, to keep a large surface area to volume ratio. 

However, you are expected to be able to calculate the SA:V ratio for a cube, cuboid or cylinder and explain how the increasing size of an organism affects the SA:V ratio.