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October 5, 2011 8:20 AM   Subscribe

If an average-sized human body were rolled into a sphere, how big would that sphere be?

Like, if you had a human-sized-and-shaped clay sculpture, how big a ball would that clay make?
posted by shakespeherian to Science & Nature (11 answers total) 7 users marked this as a favorite
Those are two different questions! Clay is not as squishy as people are.
posted by Jairus at 8:23 AM on October 5, 2011 [2 favorites]

The human body is almost the same density as water so if you can find some stats on average weight, you should be able to work this out.

@Jairus: Volume isn't affected by density so the answer to both questions is the same.
posted by VTX at 8:28 AM on October 5, 2011

You're asking for the volume of the average human body. Assuming that the average density of the human body is roughly 1 kg/L (same as that of water), and average weight in the US is about 80 kg, so you're looking for a sphere that's 80 L in volume. Circumference or the sphere = 180 cm.
posted by supercres at 8:31 AM on October 5, 2011

According to wolfram alpha, human beings are about 66 L, and would be a sphere about 10 inches in radius.
posted by apathy0o0 at 8:32 AM on October 5, 2011 [1 favorite]

From here, it looks like the density of your average human is about 0.001 kg/cm^3. The total volume in cubic centimeters of your average human, then, is M(ass)*1000. Volume of a sphere is 4/3*pi*r^3.

The radius of a sphere with that volume is solved algebraically. M*1000=4/3*pi*r^3. Radius of that sphere is rt^3(cubed root) (750*M/pi).

I weigh about 150 lbs, or about 68 kg. That makes me a sphere of about 25 cm radius, or about 20 inches diameter.
posted by backseatpilot at 8:33 AM on October 5, 2011

Jeez, I can't type. 168 cm in circumference, though that's surely too many significant digits.
posted by supercres at 8:33 AM on October 5, 2011

This is a pretty simple volume equation. V = 4/3πr^3 is the volume of a sphere. FWIW, I found a non-sourced statement amongst the interwebs that the average volume of a human is around 20.95 US gallons.

20.95 US gallons is around 79,287 cubic centimeters.

79287 cm^3 = 4/3 π r^3
r^3 = 18928.4 cm^3
r = 26.65 cm

Double to get the diameter, so a sphere 53.3cm in diameter, or 20.98 inches, give or take some lazy math skills.
posted by Mister Fabulous at 8:35 AM on October 5, 2011

wolfram alpha can do it in one go!

posted by tehloki at 8:55 AM on October 5, 2011 [8 favorites]

Thanks everyone! I am terrible at science and units and things so.
posted by shakespeherian at 9:03 AM on October 5, 2011

wolfram alpha can do it in one go!

Apparently if you smush up a human into a ball, it ends up being a sort of yellow-orange colour. Who knew?
posted by Johnny Assay at 10:07 AM on October 5, 2011

Volume isn't affected by density so the answer to both questions is the same.

It depends how determinedly you're squishing the person. The human body has a load of air spaces -- lungs, bronchi, trachea, mouth, digestive system, sinuses -- that all help to bring the mean density of the body down. You can feel this in the pool, when your chest tends to float (air in the lungs gives your thorax a lowish mean density) and your legs sink (solid muscle & bone, giving a highish mean density).

So if you're using standard-issue Giant Spiked ClawsTM to crush your victims, the air will be pushed out of your sphere'o'meat, allowing the gaps to close and therefore raising the density/decreasing the volume. Alternatively, if you can lure your adversary into a terribly clever mechanism that forms a close, airtight fit and starts constricting into a perfect sphere while you look on cackling madly at the sheer joy of it all, the air will be kept and thus the density and full volume maintained.

The eventual difference in radius would be tiny (the volume change would be small to start with, and radius scales with the cube root of the volume), but these things are important.

I'm going to assume that the body contains 38ml/kg of gas, based on the graph in this wiki page, plus a bit more to include the other air gaps.

So the calculation becomes:
Mass = 85kg
Volume including air gaps = 1litre/kg * 85kg = 85litres
Volume of air gaps = 0.0038litre/kg * 85kg = 0.323litres
Volume with air squelched away = 85 litres-0.323litres = 84.677litres = 8467700cm3

Radius with air squelched away = cube_root(8467700/((4/3)*pi)) = 27.241cm
For comparison, radius with air included = cube_root(85000000/((4/3)*pi) = 27.276cm

For these sorts of differences to matter, of course, you're going to need a much better idea of the mass and density of your candidate, and how full his lungs are. If there has been a lot of screaming, you'd probably want to knock the volume of gas down to 15 or 20ml/kg. If you're dealing with a professional, content with a few sardonic comments, you're good to go. To avoid the whole problem, why not just discern his volume by measuring the amount of water displace while he's splashing around in the pirhana tank?

As a side note, inspired by this picture, I once estimated that a sphere made up of all currently living humans would have a radius of 940.43m. So a sphere of all humanity would be almost exactly one thousandth of the length of the UK or, on that image, about one hundredth of a single pixel. The stuff about taking gas pockets into account hadn't occurred to me back then, so factoring that in is left as an exercise for MI6.
posted by metaBugs at 11:48 AM on October 5, 2011 [12 favorites]

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