Becca Tanner

The best way to grasp the concept of Fermi Problems is to give one a go. So where’s a better place to start than calculating the mass of all the oceans on Planet Earth? Of course we know straight away it’s going to be a big answer, but how big? Believe it or not, this is a relatively straightforward problem, using formulae you probably learnt in school science lessons. Before we get started however, there are some useful hints to remember so that we can successfully solve any Fermi Problem:

1. Work in powers of ten. This one is super important as the key to Fermi Problems is to make estimates rather than to use precise values, and working in powers of ten will make our lives a lot easier.
2. Guess numbers. Sometimes you’ll need to guesstimate what a value might be, for example the population of a country or (in our case) the average depth of the ocean. As long as you’re within the correct order of magnitude a guess is fine – and if you’re really not sure then google it!
3. Simplify the problem where possible. This is important when facing geometrically complicated problems such as the depth of the ocean. Of course we know in reality that the ocean floor is irregular and varies dramatically in depth, but we’re going to treat it almost like a box with an average depth and approximated surface area. 
4. Don’t sweat the small stuff. This is the golden rule of Fermi Problems as the whole idea is to keep things as simple as possible. By ignoring the less significant aspects of a problem you are obviously not going to get the correct answer, but their insignificance means that they can be happily ignored and will still leave you with a close-enough approximation at the end.

Okay, let’s do this…

First, let’s think about the equation we’re going to need. As mentioned previously, it’s a pretty straightforward one:

The density of seawater is just over 1000 kg/m³ but since we want to work in powers of ten this becomes:

So, we have the density, but we want to calculate the mass. Clearly we therefore need to work out the volume of the ocean. This can be approximated by multiplying the average ocean depth, by the surface area of Earth covered by oceans:

Now it’s time to start making some approximations. As we’ve said before, the depth of the ocean varies considerably across the planet, but its average depth is somewhere on the order of 4000m (4 x 10³ m).

The surface area of the ocean is slightly less straightforward. First, we know that the ocean covers about 70% of Earth’s surface. We also know that the Earth approximately resembles a sphere, so the surface area of the Earth can be calculated using the formula for the surface area of a sphere:

This means to calculate the surface area we need to know Earth’s radius. We can approximate this to be about 6000 km (6 x 10⁶ m), which gives us a surface area for the Earth of around 4.5×10¹⁴ m².

Now considering that the ocean only covers 70% of Earth, this value can be multiplied by 0.7 to attain the surface area of Earth’s oceans:

We’re almost there! We have so far worked out the approximate surface area of Earth’s oceans and we have an average depth of 4000m (4 x 10³ m). Multiplying these give us an approximate volume:

Now that we have our volume we can go back to our original equation to calculate the mass:

There we have it, the ocean has a mass of approximately 10²¹ kg. Better yet, the actual values currently accepted vary between 1.3 to 1.4 x 10²¹ kg, so our answer is well within the correct order of magnitude – and took all of a few minutes to calculate!

Of course, Fermi Problems can become a lot more complicated in nature – a prime example being the Drake Equation which attempts to calculate the number of intelligent extraterrestrial civilizations in the Milky Way galaxy. In the final article of this series we will look further into this infamous equation, before delving deeper into the mathematics behind the success (and sometimes failure) of Fermi Problems.  

If you would like to have a go at some more Fermi Problems yourself here is one that nicely follows on from our problem above:

If you used all the kettles in the UK, how long would it take to boil the entire ocean?

A solution video from Tom will be coming soon!

Fermi Problems Part 1: Envelopes at the Ready!

Fermi Problems Part 3: Where is everybody?