# 1 Billion Gallons of Water

A friend mentioned that the diving events in the current (Glasgow)
Commonwealth games are being held here at Edinburgh’s Commonwealth Pool.
She was right:

(That’s a screenshot from Glasgow 2014, which, with luck, will change soon.)

But what’s this “1 billion gallons of water”? Surely that can’t be the amount of water in the pool, can it? I mean, that’s *obviously* wrong. Isn’t it? It is obviously wrong. But let’s just double check.

As we all know, an Olympic/Commonwealth pool is 50m by 25m. And the depth is something like 2m on average. So the volume is 50m x 25m x 2m = 2,500 cubic metres. That still doesn’t seem like it’s going to turn into a billion gallons, but let’s carry on. A cubic metre is 1,000 litres (a cube of side 10cm has a volume of 1 litre). So the pool has about 2.5 million litres of water. And a gallon is *more* than a litre. So QED.

Trying slightly harder (and still not looking anything up) a gallon is 8 pints and there are about 1.75 (imperial) pints in a litre (or about 2, if we only care about orders of magnitude). So we need to divide that 2.5 million litres by something between 4 and 5 (8/1.75 = 4.57) which will give us something a little over half a million gallons.

So there not only aren’t a billion litres of water in the pool: there aren’t even a *million* by my estimate. The claim is out by 3 orders of magnitude.

Now, of course, if you want to defend Glasgow 2014, they don’t actually say that the pool’s volume is a billion gallons. They just stick the phrase “1 billion gallons of water” under “Royal Commonwealth Pool”. Maybe it’s the amount of water the pool will use over its lifetime. Maybe it’s just an impressive amount of water. Who knows.

Why do I care? Only because I think it’s important people develop a good
sense of scale, and can spot when numbers are “obviously” wrong (even if
sometimes “obviously” wrong numbers turn out to be correct). An excellent
discipline physicists are often taught is

meaning that whenever you’re calculating anything you should first
guess the answer (literally, just guess); then make a crude estimate
by approximating the key components of the formula; then calculate the
precise answer. The idea is that this leads you to first to develop
better intuition about orders-of-magnitude sizes (by seeing when your
guess is significantly off), but also helps you to avoid believing
calculations that are orders of magnitude off, because the guessing
and estimation stages lead you to be surprised by an answer that
seems as if it can’t be right.

Ordinary people, quite reasonably, have no feeling at all for what a billion is,
because in ordinary life you rarely if ever encounter such numbers in contexts
where they can really be appreciated. Sure, people might know that the
population of the Earth is about 7 billion, but you can’t see 7 billion
people. Even more, people will hear about states and companies spending
and earning quantities of currency measured in billions (or occasionally
even trillions), but again, those are just abstract numbers. Even when
there’s hyperinflation and prices begin to measured in millions or billions,
the appreciation doesn’t really increase, because the million or billion
just becomes a suffix.

If you don’t have a feel for a billion (and who really does?), my two
favourite ways of getting some kind of handle on it are the following:

I always remember the number of seconds in a year as “pi times ten to the
seventh”, i.e. about 31 million. Since pi squared is ten (roughly), this means
that a billion seconds is about 31 years.

The other way I like to think about it is with respect to
centimetre cubes and metre cubes. When I was at school, we had wooden
1cm cubes that you could assemble into larger volumes. Obviously, a
1-metre cube contains one *million* centimetre cubes (100 ⨉ 100
⨉ 100 = 1,000,000). If you don’t immediately get a sense of
just how big that means a million is, get hold of some of those 1cm
cubes, and start arranging them to make the bottom layer of the
1-metre cube. (You’ll need 10,000 of them.)

A billion centimetre cubes is either one thousand of these
metre cubes, or a single cube of side 10 metres. Now a 10m by 10m room
is a pretty decent size, but will typically only be 2–4 m tall. So a
billion centimetre cubes would fill a room (say) 50m by 10m by 2m tall.

And so we come full circle. Because that 50m x 10m x 2m is not so far
off the size of an Olympic swimming pool (too narrow, at 10m, but pretty
similar otherwise). And it only contains a billion centimetre cubes.
So even if you have no real sense of how big a gallon is, you probably
know it’s orders of magnitude bigger than a 1cm cube.

Labels: estimation, fermi, order of magnitude