# How sound is a non-round pound?

28 March 2017

As the Royal Mint introduces its new pound coin, Professor Simon Cox and Dr Daniel Burgarth from Department of Mathematics discuss the geometry of coins.

“We are all familiar with the 20p and 50p piece coins. They both have seven sides, but have you ever wondered why?

Of course, the coins are not seven-sided polygons (heptagons) with straight sides and sharp corners.

This is an important clue! Instead the sides are curved. Is it just to stop them making a hole in your pocket?

No, there's another reason: the sides are curved in such a way that however you measure the coin's width, i.e. the length of a line joining opposite sides of the coin, you get the same number (21.4 mm for the 20p and 27.3 mm for the 50p).

A little school geometry shows that this is only possible if the coin has an odd number of sides (see box below). So the 20p and 50p coins have constant width, a property intuitively associated only with the circle.

If you visit Aberystwyth pier you can see why this might be a useful property for a coin to have. However a coin slides into a slot machine, the machine can accurately detect its size and check that it has the correct value.

On the other hand, this wouldn't work for a coin with an even number of sides.

There is no way to curve the sides of a polygon with an even number of sides to ensure that its width is the same in every direction.

Today, Tuesday 28 March 2017, the Royal Mint will introduce a new pound coin. What is special about it? It has twelve sides!

Is this the end of amusement arcades? Will launderettes grind to a halt...?

The answer is probably not: the Mint has consulted widely, and a twelve-sided polygon is quite close to a circle anyway.

They expect that the tolerance built in to current machines to allow them to accept slightly deformed pound coins is probably sufficient to cope with the new coins.

And the corners will be slightly rounded so that they don't make a hole in your pockets.

So life should continue as normal, with the added advantage that the new coin is more difficult to counterfeit.

If you remember the twelve-sided threepenny bit, or three pence coin, produced until 1970, then you will say that we've been here before (and survived!).

But did you know that in the Bahamas they have gone the other way, and minted a three-sided coin with constant width?

For its size, it uses less precious metal than any other “round coin”, perhaps important in these times of austerity.

So the next time you open your wallet, that's a bit of geometry that you're spending.