# How do I calculate the volume of a given quantity of coins?

## Question

A rectangular chest is filled to the brim with 1,000 unevenly piled gold coins, each weighing one third of an ounce. How big is that chest? Is there a formula I can use for routinely calculating the volume of a given quantity of coins?

## Context

Some of the players in my group are, bless their hearts, quite detail oriented. As their DM, I love this, but it does mean I have to get certain things right. For my own satisfaction, and for the sake of creating a plausible world, I like to know this kind of information. Yes, I could say "the chest looks large enough to fit 1,000 coins" but some of my players are the type of people who will still want to know the size, and I want to honor their commitment to my world with good information.

## Resources

We are playing 5e. Please feel free to fill in any information missing from 5e with info from previous editions, however 5e takes preemption wherever there might be conflict. If no coin dimension is described RAW in any previous edition, you may assume any reasonable dimension of a round coin that makes sense for the given weight. For calculating packing density of coins and other inputs that might not be provided RAW, feel free to use real-world figures.

## Reverse engineer it from known metal densities (and marvel at how tiny and dense the currency is)

Given we know the standard weight of all currency as described in 5e D&D (0.32 oz or 9.07 grams, as there are 50 coins to a pound regardless of denomination), we can reference the metal density to find out what the volume of coinage is, and therefore work out how much space a given value of coins must occupy.

• Copper - 0.324 lb/in3 - ~0.062 in3/coin - 27,993.6 coins per cubic foot
• Silver - 0.379 lb/in3 - ~0.053 in3/coin - 32,745.6 coins per cubic foot
• Gold - 0.698 lb/in3 - ~0.028 in3/coin - 60,307.2 coins per cubic foot
• Platinum - 0.775 lb/in3 - ~0.026 in3/coin - 66,960 coins per cubic foot

Since pure trade bars and the actual currency have the same value per weight, I am assuming that the coins are pure and have the same densities (but, as explored here, this is not necessarily consistent with how currency has been described or depicted in D&D's history). The values given here also don't account for the fact that the coins are probably not shaped so that they perfectly tessellate. As this blog post from dmsworkshop.com helpfully summarises, assuming round coins that are about 1/16th of an inch thick and roughly 1 inch in diameter (varies depending on denomination):

To save you some math, the ideal packing density of coins is 78.6% (if neatly ordered in stacks) or about 60% if loose. This means that an unorganized heap of coins (like those stuffed into a sack) will contain about 40% empty space.

Non-round shapes would improve the packing density to varying degrees depending on the exact shape of the coin, but using round coins seems a sensible default. So taking that into consideration:

• Copper - ~22,000 coins per cubic foot (neatly stacked) - ~16,800 coins per cubic foot (loose jumble)
• Silver - ~25,740 coins per cubic foot (neatly stacked) - ~19,650 coins per cubic foot (loose jumble)
• Gold - ~47,400 coins per cubic foot (neatly stacked) - ~36,180 coins per cubic foot (loose jumble)
• Platinum - ~52,630 coins per cubic foot (neatly stacked) - ~40,180 coins per cubic foot (loose jumble)

Using these values we can calculate that, for instance, a coffer filled to the brim with 1,000 neatly stacked gold coins would be slightly over 36 cubic inches in capacity (for instance, 6 inches × 6 in. × 1 in. internal dimensions), or loosely piled somewhat less densely at about 48 cubic inches (for instance, 6 in. × 6 in. × 1.33 in.).

You're probably noticing that those are very small dimensions. Metals, especially precious metals, are dense, and sensibly shaped coins made of these metals are small - much smaller than we are apt to imagine them (see Brythan's answer for a helpful comparison to US coinage). Even if you use a chest loosely filled with 10,000sp (equivalent value to 1,000gp) that only comes to about 880 cubic inches - roughly 1 ft. × 1 ft. × 6 in. internally, but weighing in at a hefty 200 lbs!

For people accustomed to metric, here are the numbers from this answer recalculated:

• Copper - 8.96 g/cm3 - ~0.988 cm3/coin - 1010 coins per liter
• Silver - 10.49 g/cm3 - ~.8648 cm3/coin - 1156 coins per liter
• Gold - 19.32 g/cm3 - ~.4696 cm3/coin - 2130 coins per liter
• Platinum - 21.45 g/cm3 - ~.4229 cm3/coin - 2364 coins per liter

Density numbers from Density of metals, except platinum.

Fifty coins mass one pound. So one coin is .32 oz or 9.071847 grams.

Using the same 78% stacked and 60% jumbled estimates:

• Copper - 790 coins per liter stacked; 607 coins per liter loose.
• Silver - 901.9 coins per liter stacked; 693.8 coins per liter loose.
• Gold - 1661 coins per liter stacked; 1278 coins per liter loose.
• Platinum - 1844 coins per liter stacked; 1418 coins per liter loose.

Just for comparison, here are the sizes of some United States coins.

• Penny: 19.05 mm diameter; 1.52 mm thickness; .433 cm3.
• Nickel: 21.21 mm diameter; 1.95 mm thickness; .689 cm3.
• Dime: 17.91 mm diameter; 1.35 mm thickness; .340 cm3.
• Quarter: 24.26 mm diameter; 1.75 mm thickness; .809 cm3.
• Half dollar: 30.61 mm diameter; 2.15 mm thickness; 1.58 cm3.
• Dollar: 26.49 mm diameter; 2 mm thickness; 1.10 cm3.

Volume calculated from diameter and thickness.

The penny is closest in size to the gold (and platinum) coin. So if you wanted a visual, you could go to a US bank and get twenty rolls of pennies for $10. That would be a little smaller than the gold coins would be. Perhaps if you leave them in the paper, the size might be closer. Then find a box (chest) large enough to hold the pennies. Perhaps a jewelry box with a lock. The quarter is more in line with the size of silver coins. Note that there are only forty quarters in a roll, so you would need twenty-five rolls ($250) to make a thousand coins.

The dollar coin is perhaps closest to the size of copper coins. At twenty-five coins per roll, you would need forty rolls (\$1000) to make a thousand coins.

• The irony of US currency in an answer about converting to metric is great. +1 and thanks for being useful and saving me some work. – linksassin May 15 '19 at 7:43
• I was considering including metric values in my answer as well (you can see I started with a mention of coin mass in grams...) but figured as D&D already uses imperial measurements for everything, that should be understandable enough. Plus I'm lazy. The comparison to actual coin sizes is a very useful aid! – Carcer May 15 '19 at 10:16