To keep the math relatively simple and to avoid complications like integrating the partial volume of a sphere, we can approximate Smaug’s bed of gold and silver to be a cone, with a radius of 9.6 feet (1/2 the diameter) and a height of 7 feet (assuming the weight of the dragon will smush down the point of the cone by about a foot).
Now we can calculate the volume of Smaug’s treasure mound:
V= 1/3 ? r2 h = 1/3 * ? * 9.62 * 7 = 675.6 cubic feet
But, obviously, the mound isn’t solid gold and silver. We know it has a “great two-handled cups” in it – one of which Bilbo steals – and probably human remains, not to mention the air space between the coins. Let’s assume that the mound is 30% air and bones. That makes the volume of the hoard that is pure gold and silver coins 472.9 cubic feet.
We know that Bilbo eventually takes his cut of the treasure in two small-chests, one filled with gold and the other filled with silver, so it seems safe to assume that the hoard is approximately ½ gold and ½ silver, or 236.4 cubic feet of each metal.
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