If you tied a string around the earth.... [Intuition and Science Question/Riddles]

Pick one. No cheating.

[SRC: NSTA: Uncovering Student Ideas in Physical Science, Keeley & Harrington]


Are we restricted to choices a-e or are we allowed to reject them and suggest something single celled or even virus like?

I understand why you would want to go smaller but the correct answer is there!

Yep, d and r should increase proportionally to the circumfrence so I’ll go along with Tim.

Full adult, standard size? Either a cat or a goat, if the goat isn’t huge within goats.

It probably isn’t fair for math teachers to weigh in here; but I do have to say, I like the way this problem was stated! I think I’ll use that in the future. It’s better than the other way I’ve sometimes posed it to students: If you walked all the way around the world (my students can walk on water obviously!) … how much farther would your head go than your feet?

All I will say is this … remember your high school simple geometry formulas, people! You really don’t need a calculator for this one - AND - you don’t even need to know how big the earth is!

Treat it like a Fermi problem (meaning using simplified estimates in your head just to get to the rough magnitude of things - like just using three for pi because that gets you close enough.)

Or to assume most people know what a fermi problem is :joy:


Enrico Fermi was a famous physicist. But apparently he was also known for posing math problems that a person could solve with no calculator handy - or not even an envelope and pencil. Just head only - the goal being to at least get to the right magnitude of number or get your decimals in the right places.

So if I wanted to guestimate a reasonable approximation for how many calories were in the homemade peanut butter ball I just ate, I might work it thus (without looking up anything) … So I’ll guess it was about 30 grams (I know it was way more than 3g, and couldn’t have been close to 300 - since there’s no way three of them are anything close to as heavy as a kilogram). So already I’m only ball parking it here by calling it 30. And I also think I remember that every gram of carbs is about 4 food Calories and every gram of fats about 9. Since those balls had mostly peanut butter, oats, and honey as their primary ingredients, I’ll split the difference between sugars and fats and say roughly 7 Calories per gram - meaning I probably just had around a 200 Calorie snack.

Unless I was egregiously off on something there, then my answer should be reasonable. As long as the actual answer is closer in magnitude to 200 Cal than to 20 or 2000, then I was basically successful.


I don’t know why but questions like “how many golf balls fit on a school bus” interest me.

The Yale Physics Olympics gives a fermi quiz every year if you are ever bored:



Dang! I got it. D. Does depend on the goat breed though.
I haven’t comepletely lost all reason.

Go with a beautiful little angora goat. You also get the lovely fiber and a fantastic warm sweater or socks or mittens.

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  1. I did! At least the easiest ones, repeated over many classes and used in the kitchen.

  2. I didn’t see that until you mentioned it. But there’s a reason the problem adds 6 meters. It did come from a text book, didn’t it?

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6.28 would work better but also make the problem odd/worse for most people. But yes, it is from a book.

They also write: “Because of the sophisticated nature of this probe, it is best used with middle or high school students who have the procedural and conceptual mathematical knowledge to per- form and understand the calculation. If stu- dents are unsure of the height of the animals, you might consider providing height data such as the following: ant is 1 mm or .001 m; mouse is 3 cm or .03 m; cat is 30 cm or .3 m; goat is 90 cm or .9 m; and horse is 2 m. Another alternative is to provide only two choices—the ant and the goat.”

I teach in the city. Most of my students will not generally not know about the many types and sizes of goats. They don’t get a lot of nature experience. I remember polling my students one year and a huge number of them have never gone fishing before. A goat will be somewhere in between a cat and horse to them.

The expectation is that most people will pick an ant because the earth is so big, its thought adding on 6m of string couldn’t possibly do much.

The gist is that it doesn’t. The earth has a radius of ~6.4 million meters. Adding 1m to it is not very significant but that is what 6m of extra string will do.

“The best answer is D: goat. The answer is highly counterintuitive. Even after performing the calculation, many students will still not believe the answer.” – NSTA


I do this for stuff all the time, I just thought it was called being bad at maths… :joy:


I just did the math for circum’s using 6.28. With a calculator.

I love that the problem (and hopefully the discussion that would follow (maybe many times) is intended to help kids connect the math and the physical concepts.
I have said before that my physics teacher expected us to take more on faith than my pastor. You either saw the connection or you didn’t. The kids that saw it were so much more interesting to him than those who didn’t. To the rest of us: trust me.

Why is d counterintuitive?

Quite the opposite actually! It’s called: being able to recognize that when a pencil is said or calculated to have 0.2 kg mass, that a mistake must have been made because there’s no way that five typical pencils are as heavy as a kilogram! Students need to have some concept of what a plausible sounding answer should be. Too often they don’t.

So it’s a good thing indeed if your fermi skills are well-honed!


I think it’s because our minds are conditioned to think that any problems involving huge things (like our whole planet!) - must also then result in extreme answers (either very BIG - as in - how much does the earth weigh? … or very tiny - as in how much does the ocean rise if I go for a swim in it?) We’re surprised when an operation on something so big ends up giving us some very ordinary change.


So true. This reminds me of the time Neil deGrasse Tyson blew my mind when he said the Earth in the hand of a giant would be smoother than a cue ball.

Because the earth is 24,000 miles in circumference. Most of my students would expect the string to not do much with only 6m added to that distance. Many of them would also not think to do a circumference calculation at least at first.

Not smoother than a professional grade cue-ball made for a tournament from what I remember but yes, as smooth as a well used one.

I do this demonstration with my class where we model out the earth on the floor. Its about 8,000mi in diameter so I work with the scale of 1 mi = 1mm. I refresh their memory on a meter stick (100 cm and 1000mm per stick) and it ends up that we have to lay 8 meter sticks on the ground to model the earth (8 meter sticks = 800cm = 8000 mm and 8000mm = 8000 mi on our scale).

We then draw a giant 8m circle (I had trouble erasing the dry erase from my floor this year!) on the floor. representing the earth. I ask students to draw a dot showing me how deep the oceans are and how tall the atmosphere is after that. The oceans have an average depth of 2mi so the correct answer is 2mm below the surface.

Technically I suppose the atmosphere never ends but:
~50% of mass of atmosphere lies below 3.5mi (= 3.5mm)
~90% is below 10 miles which his only 10mm or 1 cm.
~ 99% us below 20 miles (= 2cm)
~99.99997% lies below 60 miles (=6cm)

Seeing this modeled out live in front of them is always an aha moment.

I then mention the billiard ball thing. And how astronauts often point out the thin sliver of our atmosphere. I tell them the oceans are like puddles on the earth and the atmosphere like plastic wrap around a basketball or the skin on an apple.

I might just buy a 30ft by 30ft blue tarp and cut it into an 8m circle to preserve my classroom floor.

But its a good way of giving a sense of scale to the earth, a good way to talk about pollution and how finite our ocean and atmosphere is and also a good illustration of why earth is an inner ROCKY planet. Students think the oceans are really deep but usually don’t realize that the while the deepest part is 7 miles, it is about 4,000mi to the center of the earth.



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