There are certain mathematical and scientific problems that require guestimation, that is; educated guesses and estimates. These sorts of problems cover a wide range, from working out the number of coins or sweets in a jar, to working out the number of grains of sand held in a beach, to working out how many individual raindrops have fallen to the Earth’s surface since its formation.

They are know as Fermi Problems, and are a topic my uncle goes into a lot of detail on his blog: mathsevangelist.

To take the number of raindrops that have fallen to Earth in all of its life as an example, my uncle provided a bit of the research necessary to work this one out:

Useful facts:

– Raindrops vary between 0.1mm and 9mm, but are typically, say, 1mm in diameter.

– World average annual rainfall is 990mm i.e. very close to 1m.

– World surface area is 510,072,200km² or about 5×10⁸km² = 5×10¹⁴m².

To add to this, water has been falling from the sky for at least 4.4 giga-years ago¹.

Given this information, can you figure out how many raindrops have fallen to Earth since its formation?

With an annual rainfall of ≈1m, the volume of water that falls to Earth each year is ≈5×10¹⁴m³.

And if each raindrop has a volume of , then the number of raindrops that fall each year is:

But that’s only for one year!

For the whole of Earth’s lifetime, we must multiply that by its age, or at least, when rain began to fall—4400 million years ago. That gives:

That’s roughly 4,200,000,000,000,000,000,000,000,000,000,000 or 4.2 thousand quintillion! (4.2 American decillion¹)