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.
Best Answer
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.
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):
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:
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!