## How Big Is A Coin?

The weight of individual coins has varied from edition to edition, as has the relative value of the various coins. To my recollection, no edition has actually pinned down a specific size though. Even if a size were specified somewhere, I would suspect it was a purely arbitrary number. Most folks are fine just knowing that a coin is “coin-size” and leave it at that. But I’m not “most” people. :)

I would have sworn I had already written an article on this but when I went to find it, to use as a link in an upcoming article on gems, I couldn’t locate it. I guess that’s just one of the “perks” of getting older. <sigh>

### 50 Coins to a Pound

Various weights have been suggested over the years. I prefer to use 1/50th of a pound per coin. If you use a different weight, it shouldn’t be too difficult to adjustment accordingly.

### US Coins

I thought it would be useful to start off looking at some actual coins that most people are familiar with. The data in the following table was taken from Wikipedia:

Coin | Diameter | Thickness | Weight |
---|---|---|---|

Penny | 0.750″ | 0.061″ | 2.464g |

Nickle | 0.835″ | 0.077″ | 5.000g |

Dime | 0.705″ | 0.053″ | 2.268g |

Quarter | 0.955″ | 0.069″ | 5.670g |

Fifty Cent Piece | 1.205″ | 0.085″ | 11.340g |

Sacagawea Dollar | 1.040″ | 0.099″ | 8.100g |

Silver Dollar | 1.500″ | 0.102″ | 22.68g |

Most people agree that the size of a fantasy coin should be somewhere between a nickel and a fifty cent piece, with a quarter being more or less in the middle of that range. I like the size of the quarter so I’ll start off testing some numbers around that size.

### Specific Gravity

Weight (in grams) divided by volume (in cm^3) gives us the specific gravity (in g/cm^3), a representation of how dense a material is. We already know the weight: 50 coins to a pound (0.02 pounds or 8.96g). The following chart gives the specific gravity of the various precious metals.

Copper | 8.92 g/cm^3 |

Silver | 10.50 g/cm^3 |

Gold | 19.30 g/cm^3 |

Platinum | 21.45 g/cm^3 |

So all we need to do is work the equation backwards (specific gravity times weight equals volume) and determine what dimensions we want to use that will provide the necessary volume.

However, minted coins (as well as jewelry, statuettes, and other works of art made from precious metals) are not made from pure copper, silver, gold, or platinum. The metal will also contain impurities that will alter the specific gravity of the alloy. This may be due to naturally-occurring impurities that were not extracted, or other metals that were intentionally added to make the metal easier to work with.

Furthermore, I want all coins to be of the same size and of the same weight. To accomplish this, all precious metals must have the same specific density. To that end, I will assume that each of the precious metals has other metals added to them that result in each alloy having the same specific gravity. This isn’t too great a stretch and should still remain within the realm of possibility.

This also allows us some latitude in what specific gravity to use.

### Determining an Exact Size

I stated above that I would like to use a quarter as a model. A US quarter has a diameter of 0.955″ (2.4257cm) and a thickness of 0.069″ (0.17526cm). The volume is then 3.14159 * (2.4257/2)^2 * 0.17526 = 0.809928 cm^3. Multiply that by 8.96 grams (the equivalent of 50 coins to a pound) and you get a specific gravity of 11.06271 g/cm^3. That value falls nicely within the middle of the range of specific gravities of the precious metals involved (almost like I planned that in advance).

The specific gravity of an actual quarter is 7 g/cm^3 so a fantasy coin would weigh 60% more than a US quarter.

Note that the dimensions of a quarter (used above) are very nearly (but not exactly) 15/16″ diameter and 1/16″ thick. Using these dimensions instead would change the volume from 0.8 to 0.7 cm^3, and result in a specific gravity of 12.67345 g/cm^3 (which also would be a reasonable value to use). I will stick with the actual dimensions though.

### Who Cares

Offhand, I can’t think of a single reason for wanting to know the exact dimensions of a coin. But I did want to know, with at least reasonable accuracy, the rough dimensions so I could gauge how many coins could fit in a sack or chest. Some people envision a gold piece as being the size of a nickel while others maintain that a gold piece is enormously bigger (say the size of a silver dollar). That’s a pretty huge range and I wanted some data to show those folks that would back up my choice of a gold piece being the size of a quarter.

### Filling a Chest

Now that we know the size of a coin, it’s time to determine how many coins fit in a chest. I define a small chest as having inner dimensions of 9″ x 6″ x 4″ which composes a volume of 216 cubic inches. A single coin has a volume of 0.049425 cubic inches (0.8 cm^3). If you were to melt down the coins and pour them into the chest, it could hold 4,370 coins (216/0.049425). That sets our upper limit. Since a bunch of coins dumped into a chest will not fully fill the area within the chest, the number of coins the chest can actually hold will be far fewer. Sadly, I don’t know how to accurately determine what that number is. Therefore, I am arbitrarily going to assume that the chest can hold 3,750 coins (roughly 85% of the upper limit). This is a convenient number as it means that the chest will then weigh 75 pounds plus the weight of the chest itself.

**Edit:** Some preliminary research shows that stacking coins in rows and columns results in 21.46% wasted space (1 – pi*r^2 / d^2). Stacking coins in offset rows results in 13.59% wasted space (higher at the edges). Loose coins will most likely come closer to 30% wasted space, resulting in a maximum of roughly 3,000 coins per chest (60 pounds place the weight of the chest).

### Bars of Gold

When vast amounts of gold or silver are involved, it may be more convenient to deal in bars instead of coins (I’ve never been fond of bank notes in a fantasy environment). One bar of gold (or copper, silver, platinum) is equivalent to 1,000 coins and weighs 20 pounds. The dimensions are roughly 10″ x 5″ x 1″ making them easy to stack and carry.

### Shaved Coins

If an unscrupulous person were to shave a tiny bit of gold from each coin (or bar), she could melt down those shavings and form them into rings and such for a nice profit. To circumvent this, coins are minted with reeded edges (as seen on the US quarter). Merchants can then easily see if a coin has been shaved and refuse to accept it. For transactions of any significant value, coins and bars are weighed to determine their exact weight.

Actually promissory notes (paper currency) in fantasy can work well, and can also be an interesting “screw with the PCs” trick as well.

Basically they should read like a cheque or similar.

“This note entitles its bearer to an amount of (insert crazy value here like 137 gold 23 silver) coins from King Whassisname of Overthereland”.

Now, the way it screws with in a possibly productive way is…

1. No common person in Thingamajigland accepts the note because it’s from Overthereland

2. Mr Moneybucks the Moneychanger will accept it, but he won’t give you full value (because it costs him to get to Overthereland and convert it into cold hard coin)

3. Mr Buyandsell will accept it but can’t give you that kind of cash immediately. If you’ll hang around for a couple of weeks he was going to go sell a shipment of XXXX in Someplace before returning here and heading to the Capital of Overthereland where the note is useful.

4. That note is very old, and King Whassisname was deposed. You might be able to get it out of one of his relatives… maybe.

I do like some promissory currency, but I feel it should always be for a strange amount rather than the “Five Pound Note” or standardised promissory currency we have today.

No doubt bank notes can create some interesting situations (and you’ve got some great examples there!) but what I don’t care for about them is that they make things too easy. Having to transport a substantial number of coins is an adventure in itself. It takes an ingenious plan to do so without attracting unwanted attention.

However, I like the ideas you’ve come up with and am starting to rethink my position. At least in certain circumstances, I can see where they could work out quite nicely.

I am still unable to locate any method of statistical analysis that will predict the number of coins that will fit in a chest when randomly dispersed. I originally estimated that loose coins would result in 15% wasted space. After calculating the wasted space resulting from various methods of stacking coins, it seems that estimate is unrealistic. I now suspect that 30% wasted space will be more accurate. I have edited the article to reflect this.

Mike,

The easiest option, is to attempt to determine how many coins you could maximally fit into the area (there are mathematical solutions for this, I’m just not sure where they are, but they would relate to hexagonal tesselations) and the minimal number (using a square tesselation). You could assume then a bell curve with these two as the maxima and minima of the statistical curve… and with some jiggery-pokery you’d be able to determine the wastage.

I might dig around for you and give you a nice range.

Found something…

Hexagonal Packing (so that the centres of the circles form the corners of a hexagon) reaches around 90% efficiency in 2-dimensions

Square Packing (so that the centres of the circles form the corners of squares) reaches around 78.5% efficiency.

So as a simple estimate, I’d suggest that you’d have around a 75-80% efficiency in 3-dimensions. Which conversely means you’d have 20-25% wasted space.

Aaaah abstract mathematics, how I love you so…

Thanks for looking that up. The figure for square packing matches my figures:

(1 – pi*r^2 / d^2) = 21.46% wasted space. However, I get 13.5879% wasted space in a hexagonal pattern (not counting edge anomalies). I’ll have to go double check my math.

I posed the question on rpg.net and (amongst all the babble) I got a reply from someone who, through some physical testing, came up with roughly 50% wasted space on random dispersement. I find that pretty hard to believe so I think I need to get a few hundred dollars worth of quarters and do some empirical testing of my own. I really expect wasted space to be around 30%, possibly even 25% in a container large enough to allow for maximum settling.

Some of those folks on rpg.net really make me wonder if math was one of the programs cut from the high schools!

I didn’t calculate those, I did a few good google searches and looked at the websites with the nice trustworthy university names… and compared them to the Wikipedia answers (Wikipedia is likely to be right on this one).

The issue isn’t so much the “space” or the “number” its how agitated they will be. If you got those coins and basically gently rattled them around on the back of a cart along a glorious unpaved road drawn by horse or oxen… you’d probably over a journey of a day or two get enough time and agitation to improve the efficiency of the arrangements. It’s a curious physics thing and stunningly things will tend towards the most efficient packing if put inside a container and agitated enough. Just like there is also a mathematical and physical reason why perfectly coiled ropes if left (or even agitated) will knot themselves. The physics leads to it being a more efficient existence for a rope/cord/cable.

If you really want to get into this deeply, go searching for Packing Theory.

Wikipedia was indeed correct. When I calculated the hexagonal packing I took [unused area/coin area] instead of [unused area/(coin area + unused area)] <hangs head in shame>. After correcting that I got the same answer wikipedia lists: 90.69% efficiency.

Unfortunately, packing theory can’t resolve the problem of random dispersement (shaken or stirred). It’s a statistical problem and esoteric to the point that I don’t expect to find a solution anywhere. Averaging out repeated experiments is the best bet. I just haven’t had the time to play with it yet.

The approach that I’ve taken with money is a bit less scientific, but I’ve tried to tie up loose ends through explanation. I’ve always found it problematic to think that the loose governmental units of a D&D-like world would create a universal currency, so the history for the world that I built has a set of reasoning behind it. Then, in the history, a few enterprising gnomes set about to create coins of uniform size and un-counterfeit-able design. In general, coins are treated as being more valuable than precious metals themselves, so players have reason to want to take any discovered metals to the minting center. I dealt with the varying weights and densities by explaining that the formula for creating the minted coins adds enough alloying to bring each coin to a uniform weight (I adopt 10 grams).

To add something for my players, I designed an image of what each coin’s face and back looks like, then ordered a large set of poker chips with the images printed on them. It adds a lot to the experience for players, especially when in-game gambling uses their actual money, which become tangible as it actually sits in front of them.

One of the things that I really enjoy about this blog is the perspective that you take on details. Ordinarily, I’d tell my players, “You open the chest and see that it’s filled with silver and gold pieces,” and when they ask how much to add to their inventories, I just throw out a number that seems reasonable to me. Avoiding giving a size to the chest was my way out. But trying to approach it from this direction is interesting, and I’m definitely going to try incorporating it into my next game.

Ok – so response to ancient post, but was perusing the site. Drawn #80 or #81 had a great article on the volume and weights of different metals for coins and how much space and weight they represented. Author of article had done a bunch of research on historical Greek, Roman, and various European coins as background.

And of course that should have been Dragon #80 or #81…