Most sources I’m seeing agree that a chicken egg shell is about 0.35 mm thick. A few (extra large) eggs I pulled from my refrigerator average about 45 mm by 60 mm. That would give a volume of about 63,620 mm[sup]3[/sup]. Subtracting the shell, the interior is then 44.65 mm by 59.65 mm, giving a volume of 62,270 mm[sup]3[/sup], so the shell is 2% of the volume of the egg.

However, Wikipedia says “the [chicken] egg shell constitutes about 13% of the weight of the egg”. Surely the shell is not 6.5 times as dense as the interior. I can’t explain this, unless the (uncited) 13% number is simply an error.

Thanks BigT and Novelty Bobble. I think you both hit the nail on the head. Once I stopped thinking about the object in the middle and just thought about the rope, it was much easier to say, “Yeah, that makes sense, now that it’s explained.”

I buy the cheapest eggs whose packaging will convince my wife I bought the super-organic cage free no-antibiotic yogi approved eggs.

The exception, weirdly, is my local True Value hardware store, whose owner clearly has a chicken farm because why else do you sell eggs in a hardware store? Eggs from Virgils in a polystyrene container are okay because they “taste better.”

Sorry I’ve wandered so far away from the OP, but, like, I’m totally into the circumference of the earth.

Then again, odds are real good Virgil’s chickens are free running and get no antibiotics. They probably haven’t been blessed by a yogi, but OTOH they have eaten lots of genuine organic bugs & grubs & such, which is where all the extra flavor in the eggs comes from.

I’m sure your wife will be glad to learn this last factoid about wholesome home grown eggs.

This thread reminded me of a video posted a few weeks ago by vsauce about “napkin rings”

Apparently, any two cored spheres (like an apple with a fat core removed from it) of the same height have the same volume…even if one of them is earth sized.

Here is another way to look at the seemingly insignificant increase in the length of the belt, in such a way that it might make it easier for you to picture how the 1 foot rise will occur…

So…remember the old pi formula for figuring a circle’s circumference once you already know the diameter.

Pi=3.1412, right?

OK…so…three-foot increase in belt? This adds three feet to the belt’s circumference.

Well, it now also equals a roughly one-foot increase in its new diameter. So…that is one foot above the old diameter that was snug to the Earth’s surface.

Just keep in mind the pi adage that a circle’s diameter is roughly one third of it’s circumference, and this whole thing comes into focus a lot better.

Yikes. Standard graph paper has what, 1/10" or 1/4" squares?
At 1/10" that’s 100 inches which is roughly do-able in an old house with 10-foot ceilings. Pretty sure he wasn’t using 1/4" graph paper. OTOH, in science class we did have 1mm graph paper - much more manageable.

So if we do the OP’s question with algebra… Assume the belt is 6 inches off the ground, with rounding (sorry)
Area of a circle - the equator- is A=(pi)r^2
Area under belt - B=(pi)(r+0.5)^2
Difference D=(pi)((r+.05)^2- r^2)
So the key is - remember (a+b)^2 = a^2 +2ab+ b^2
D= 2(pi)(r)(0.5) + (pi)(0.4)

The key here is the difference D is 2(pi)(1/2)(radius of earth) (ignoring the second term, 3/4sq ft.)
(radius of earth) is approximately (4,000miles)x(5280) feet = 21,120,000 feet
Difference is therefore that times pi, or about 66.3 million square feet. (plus 3/4 sq foot)

This illustrates that when you get into squaring - and cubing - and other exponential functions, numbers explode very quickly.

The OP being well-answered, I’ll follow up on this rabbit-trail:

How is that troubling? If you can explain your informal thinking maybe we can give you a different informal way to understand it correctly.

In lots of math, the formalisms are what they are and even non-experts can accept them as formalisms. The hard part for non-experts is chunking the formalisms into a simplified mental model that matches the underlying reality well enough. Said another way, it’s nice for things to make qualitative sense as well as quantitative.

Share your model and maybe we can give it a tune-up.

It only takes 23 people in a room for there to be a 50/50 chance that two have the same birthday?!? Out of 365 possible days? It just feels wrong. That’s all I got for my informal thinking but I’d love a third-grade-level explanation.

I have what may indeed be a stupid question, but are we imagining the belt goes around the surface of the Earth or several miles above that? If it’s on the surface, don’t you have to factor in mountains and other irregularities on the earth’s surface? Sorry–my background is in social studies, not math, so I tend to think accordingly, and this factor has always puzzled me.

When I first came across this “surprising fact”, at the age of 10, the question was posed the other way around: if you suspended a rope 1 foot above the surface of the earth, all around the equator, how much longer would it be than the earth’s circumference. When it emerged that the correct answer was about 6 feet, my reaction (which I think is the expected reaction) was surprise that it was “only” 6 feet. All those thousands of miles around the earth and it only takes an extra 6 feet of rope?