Where Would We Fit?

In the classic book 101 Puzzle Problems, Nathaniel B. Bates and Sanderson M. Smith make an astounding claim:

The volume of the 1970 world population is less than the volume of the Houston Astrodome.

That is, if you packed all of the people on Earth who were alive in 1970 — all 4 billion of them — as tightly as possible, they would’ve actually fit in the stadium, and they could’ve seen Evel Knievel jump 13 cars or Billy Jean King defeat Bobby Riggs.

The Astrodome, originally named the Harris County Domed Stadium, was dubbed the “Eighth Wonder of the World” when it opened in 1962.

Before you read any farther, you might wish to do a Fermi estimate to verify the reasonableness of this claim.

It’s an amazing fact!

Unfortunately, it’s also wrong.

Bates and Smith argue that the average human body has a volume of approximately 2 cubic feet. Their argument? That water weighs 62.5 pounds per cubic foot, and “the human body consists mostly of water.” Fair enough, if we think that an average human weighs about 125 pounds, which they probably didn’t, even in the 1970s.) Personally, I would’ve attacked the estimate a little differently. An average human can fit in a box that is 6 feet tall with a base that measures approximately 0.5 square feet — I know; I’ve been to funerals — and such a box has a volume of 3 cubic feet. But there’s a fair amount of wasted space in that box so, yeah, 2 cubic feet seems like a reasonable estimate.

Therefore, the world population in 1970 would have had a combined volume of 8,000,000,000 cubic feet.

So far, so good.

But Bates and Smith then make their statement about the Astrodome, with no calculations or estimates about the volume of the stadium to justify the claim. Please understand, I don’t fault Bates and Smith; as I understand it, the claim about everyone fitting in the Astrodome was a ubiquitous factoid during the decade of bellbottoms and disco. Still, you’d think that two mathematicians would have checked the stat before including it in a book.

The Astrodome has an outside diameter of 710 feet and a height of 208 feet. A cylinder with those dimensions would have a volume of

V = π × 355² × 208 ≈ 82,000,000 cubic feet

Although this is surely an overestimate, it’s still an order of magnitude too small to fit the world’s population in 1970. There’s not nearly enough space to store all those buggers.

Bates and Smith state that the volume of the 1970 population “takes up about one-eighteenth of a cubic mile.” And I think therein lies the error. The height of the Astrodome is just shy of one-eighteenth of a mile, and the diameter is a whole lot more than one-eighteenth of a mile, so someone (incorrectly) assumed that a container with those dimensions would easily hold 1/18 cubic mile. But that’s not right. As any geometry student will tell you, shrinking each dimension to a fraction of its original length results in a volume that is the cube of that fraction. Oops.

The diameter of the Astrodome is about 1/7 of a mile, and the height is about 1/22 of a mile, so its volume would have been less than 1/1000 of a cubic mile.

So, what container would have been big enough to hold the 1970 population?

Honestly, who cares?

Things have changed. Today, we are bumping up against 8 billion people worldwide, the Astrodome was declared “unfit for occupancy” in 2016, and instead of calculating the volume of stadia, we can simply use Google to find the volume of some really large places.

In fact, Google claims that the volume of the Astrodome is 42,000,000 cubic feet. If we can believe what we read, then the calculations above were, sadly, unnecessary.

So, you may be wondering, what places in the world could hold all of us? That is, what has a volume of 16,000,000,000 cubic feet?

The building with the greatest volume in the world is the Boeing Everett Factory. It has a volume of almost 500,000,000 cubic feet — which means you’d need 32 of them to fit today’s world population.

The manmade structure with the greatest volume is the Great Wall of China, tipping the scales at just under 20,000,000,000 cubic feet. That would be large enough, though how would you get all of those people inside? For easier packing, you could turn to nature and just dump everyone into Sydney Harbour, which has a volume of 562,000 megaliters, or about 19,500,000,000 cubic feet.