Someone once wrote that a ton of gold, if cast into a sphere, would have a diameter of 17 inches. It’s counterintuitive to think of that much mass in the volume of a small beach ball. Naturally I wanted to check up on this claim.
The first time I tried to check it, I used a crude volume conversion for water: “a pint’s a pound the world around”. Turns out it’s not all that accurate. I came up with an answer in the neighborhood of seventeen inches, but I had no faith in my math.
Math about gold tends to confound. There’s the confusing matter of the Troy Ounce, there’s the fact that “gold” is a catch-all term for an array of alloys none of which is pure in the real world, and there’s the dazzle of the subject matter itself. Here’s a picture of the world’s largest bar of gold:
I found that picture here. If you follow the link, you can see a bunch of Americans having difficulties trying to figure out how much that ingot weighs. Could you move it with a hand truck, or would you need a pallet jack or even a forklift?
Finally a consensus emerged: that bar weighs about 550-600 Imperial pounds. And once again the notion of a ton of gold fitting into a 17-inch sphere doesn’t seem quite right. So I decided to get the calculation right once and for all.
The specific question is: what would be the diameter, in inches, of a sphere consisting of pure elemental gold and weighing exactly 2000 Imperial pounds?
The problem with the Imperial system of measures is that measures of volume and distance were established separately and without reference to one another. Thus you can’t relate, say, gallons to inches without the use of a cumbersome conversion factor. To simplify matters we’ll have to take a detour trough the Metric system.
Our input is the mass: 2000 pounds. The conversion factor to kilograms is .4535, so:
2000 * .4535 = 907
Since a liter of pure water has a mass of exactly one kilogram, we can say that the Metric volume of an Imperial ton of water is 907 liters. So what’s the metric volume of an imperial ton of gold? This table gives the specific gravity of “gold, pure” as 19.32, so:
907 / 19.32 = 46.9462
Okay, now how big a ball is ~47 liters? Since I want the answer in centimeters, I’ll convert liters to cubic centimeters the Metric way: just shift the decimal point. So I’ve got 46,946.2 cubic centimeters of gold. There’s a standard formula that relates radius to volume of a sphere, which goes: “V = (4 Pi r^3) / 3”. Turning this inside out to solve for r, you get “r = CUBEROOT((3 V) / (4 Pi))”. Weird, huh? I looked it up, but then I had to solve for r myself before I was sure.
Anyway, plugging in 46,946.2 to the formula:
3 * 46,946.2 = 140,838.509317
4 * Pi = 12.566370614
140,838.509317 / 12.566370614 = 11,207.572468
CUBEROOT(11,207.572468) = 22.3788201
The radius of the sphere is 22.37… cm. Dividing by the conversion factor 2.54, we get a radius of 8.810… inches. Which gives a diameter of 17.621 inches.
Knowing this, we can maybe wrap our heads around the idea of that bar of gold up there. We can convert the Metric volume of our sphere to cubic inches simply by cubing the conversion factor:
46,946.2 / 2.54^3 = 2865
So we say a ton of pure gold is within a hair of 2865 cubic inches. We can estimate the dimensions of the bar of gold by using the woman’s hand for scale. It would appear to be a petite hand, so I assume the distance from the heel of the hand to the fingertips is seven inches. Obviously the bar is tapered from top to bottom, but we can simplify this by looking at the length and breadth dimensions halfway up the taper and pretending the thing is rectangular at those dimensions. So I’m guessing the bar is 14 inches long, 7 inches wide, and 8 inches high. That gives a volume of 784 cubic inches and a purported weight in the range of 575 pounds:
784 / 2865 = 0.274
575 / 2000 = 0.2875
Close enough! We’re not totally out to lunch here.
It’s interesting to think what would happen if you could somehow get hold of a ton of pure gold and cast it into a sphere. You’d have to store it very carefully or it would deform rapidly under its own weight. To keep it spherical you could embed it in glass (though the glass would deform too over time), keep it in cryo storage or maybe in orbit. Or you could just leave it on the floor and stop-motion film it as it turned into a pancake with cracks around the edges.