Space for 30,000 pound coins

At work I sit next to a Clyde supporter called Tony. As well as being a football club with one of the shortest names in the United Kingdom (pipped to the top spot by Bury), there is an interesting story about the club relating to a footballer whose contract was terminated early and was paid off to the tune of £30,000.

The story goes that the Clyde chairman did not like this footballer, and to spite him paid his money in pound coins. The poor guy was obliged to hire a large lorry to collect £30,000 in pound coins and transport the money to the bank.

Tony and I then decided to work out how much space exactly £30,000 in pound coins would take up. Would the guy have been able to get away with a transit van, or would he have needed a huge 40 tonne articulated lorry driven by someone from Holland?

It has been 23 years since I got an A grade for ‘A’-level maths and won the maths prize at school for my paper on prime numbers, so my memory of the tools to use to solve this problem was rather hazy.

Clearly the first step was to ascertain the dimension of a pound coin. A search on Google returned the answer: diameter: 22.5 mm, thickness: 3.15 mm and mass: 9.5 g. So the weight of 30,000 pound coins is easy: 0.0095 x 30000 = 285 kg, light enough to be carried by a transit van. I remembered that the volume of a cylinder is the area of its cross section multiplied by its length. So the volume of 30,000 pound coin is π x (0.0225 ÷ 2)² x 0.00315 x 30000 = 0.0376 cubic metres, which doesn’t sound very much. Tony and I wanted to visualise that. What would be the length of a side of a cube with a volume of 0.0376 cubic meters?

After much experimentation and scraping the maths rust from the inside of my brain it dawned on me that the answer is simply the cube root of 0.0376, or in other words 0.0376 to the power a third. This is because what the answer is, raised to the power three (the three sides of the cube multiplied by each other) would give the original volume. Doing the calculation 0.0376 to the power a third yields a side of just 33.5 cm, easily enough to fit in a transit van or, for the matter, the small boot of a sports car, have a look at this article on ford.

So to cut a long story short, the footballer in question would have been given a cube of pound coins 33.5 cm x 33.5 cm x 33.5 cm weighing 285 kg. OK, he would have needed a couple of strong mates to help him lift it, but he could probably have fitted it in the boot of his car, and certainly would not have needed to hire a lorry.

Afterwards I realised that strictly speaking the side of the cube of pound coins would be rather more, because the calculation above assumes that pound coins are melted down into a solid block, and does not take into account the gaps between the coins. Further, the side of the cube needs to be a multiple of 2.25 cm (the diameter of a pound coin)

So rather than multiply the area of the coin its thickness to get the volume, the total area (including the space around it) should be used. This, conveniently, is a square. So the volume taken up by one of 30,000 pound coins stacked into a cube is 0.0225 x 0.0225 x 0.00315= 0.0478 cubic metres, the cube root of which is 36.3 cm. This is the space for just over 16 pound coins. So the cube of coins would actually have a base of side 16 x 2.25 = 36.3 cm, containing 16 x 16 = 256 coins. A cube of 30,000 coins would have 30000 ÷ 256 = 117 complete stories, plus one more incomplete story containing 30000 – (256 x 117) = 48 coins giving a height of 118 x 0.315 = 36.9 cm. So the true dimensions of a near-cube of 30,000 pound coins are 36.3 cm x 36.3 cm x 36.9 cm, still small enough to fit in a car boot.