Giant watery balls

A reader writes:

I recently saw a news article that linked to this government page:
...which says if all Earth's water (liquid, ice, freshwater, saline) was put into a sphere it would be about 860 miles in diameter.

Now I understand an 860-mile sphere is massive, so even though that sounded small I could accept it, until they state the estimated volume of water on earth at 332.5 million cubic miles.

So how do you cram 332,500,000 cubic miles into a 860 mile sphere?


Quite easily, actually!

The volume of a sphere is four-thirds pi times the radius squared cubed [Sorry I left that error there for so long, commenters!]. So if the radius is 1 unit, the volume is 4.19 cubic units.

The radius of an 860-mile sphere is 430 miles. 430 cubed is 79,507,000. Four-thirds pi is about 4.1888. Multiply that by 79,507,000 and you get about 333,038,143, a number less than 0.2% larger than 332,500,000. The difference is accounted for by variations in precision in working out the number, since this is really only a ballpark figure and taking it to nine significant digits is silly.

To "sanity check" this if, like me, you always feel mildly nervous about the order of operations for a calculation like 4/3Πr^3, consider the volume of a cube 860 miles on a side.

The volume of a cube is of course just its edge-length cubed, and an edge length of 860 miles gives a volume of 636,056,000, a nice sane-sounding 1.91 times the volume of the sphere that'd neatly fit in that cube.

My own second-favourite way-to-visualise-the-quantity-of-something is that all the gold in the world (not including gold we have yet to dig up or somehow extract from seawater) would make a cube only 20 to 22 metres on a side, depending on who you ask. To help visualise the size of the cube, 21-ish-metres is about the length of two city buses parked nose to tail.

Because gold weighs 19.3 grams per cubic centimetre, though (11.16 ounces, or 10.16 troy ounces, per cubic inch), a 21-metre-on-a-side cube of gold would weigh 178,737 tonnes. So I suppose you wouldn't have to worry too much about someone stealing it.

(Unless you are very wealthy, you probably can't buy a large enough lump of gold - especially at today's outrageous prices - to really appreciate its density. At current prices, one kilogram of gold would cost you more than $US51,000. Tungsten, however, is 99.7% as dense as gold - I'm sure counterfeiters have gilded tungsten for profit many times - and it's much more affordable, though still expensive. The good people of RGB Research {here on eBay US, here on eBay UK, here on eBay Australia} have their one-kilo tungsten cylinders on sale again for a mere $US220 plus rather pricey delivery. If you can afford one, and have the slightest interest in science toys, I urge you to buy one; my own tungsten cylinder is one of my most treasured possessions. And one of the most durable, too; if the house burns down the tungsten cylinder, like my Bathsheba Grossman Metatrino, will be sitting intact in the ashes.)

My most-favourite way-to-visualise-the-quantity-of-something is that if you breathe on an ordinary marble, the thickness of the layer of condensation from your breath on the marble is approximately to scale with the thickness of the atmosphere on the earth.

(And another one, that doesn't really make anything much easier to understand but is prime stoned-party-talk, is that a human is about as much bigger than an atom as a galaxy is bigger than a human.)

Psycho Science is a regular feature here. Ask me your science questions, and I'll answer them. Probably.

And then commenters will, I hope, correct at least the most obvious flaws in my answer.

7 Responses to “Giant watery balls”

  1. hornetfig Says:

    "The volume of a sphere is four-thirds pi times the radius squared". Ummm... But you have it correct in algebraic formula later on (albeit with a capital Pi)

  2. klightspeed Says:

    If you go by mass, then a typical model is about as much more massive than a hydrogen atom as Sol (our sun) is more massive than that model, and as the universe is more massive than Sol.

    If you go by height, a typical model is about as much larger than an atom as Pollux is larger than that model, and as the Milky Way is larger than Pollux.

    That model is about as much larger than an atomic nucleus as the solar system is larger than that model.

    On the other hand, the Earth is about as much larger than an atom as the local supercluster of galaxies is larger than Earth.

  3. pavium Says:

    My own 'sanity check' said you can't get a volume in cubic metres with a formula which squares the radius.

    The volume of a sphere is four-thirds pi times r cubed (not squared)

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