How Much String Would You Need To Wrap The Earth?

Let us say that you are to wrap the earth with a single piece of string around the equator. This seems to be an enormous, yet simple task. However, this task is just an excuse that leads you into a mathematical puzzle that reveals an interesting aspect of human behaviour.

To start, I will be giving a brief description of this puzzle. Then, I will spend some time analysing the human behaviour associated with the puzzle. Finally, I will explain the logic behind solving it.

At the end of this article, you would ideally gain knowledge that enables you to associate this puzzle to similar real-life situations. Consequently, you will likely be able to make better decisions under such conditions.

This essay is supported by Generatebg

How Much String Does It Take to Wrap the Earth?

We start with a simple question: How much string does it take to wrap the earth (lengthwise)? Since you will need to wrap the string around the equator, this becomes a simple problem to solve. All you will need to do is consider the earth a perfect sphere and measure its circumference at the equator. Then, the value of this circumference would be the length of the string that is needed to wrap the earth.

For ease of visualization, let us say that the Earth’s circumference at the equator is 40,000 Kilometres (Km). So, the string would need to be 40,000 Km long. Unfortunately, you get no prizes for getting the answers right so far (if you were indeed answering). That was just a warmup.

How Much More String Does It Take to Wrap the Earth?

After wrapping the earth, suppose that I require you to wrap the earth again with another piece of string. But this time around, the second string has to hover at a height of 1 centimetre (cm) from the earth’s surface. Again, we assume that the earth is a perfect sphere for this puzzle.

The question now becomes: How much longer does the second string need to be compared to the first string in order to hover 1 cm above the earth’s surface? You need not answer this precisely. If you have a rough guess or idea about the number, hold it at the back of your mind.

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A Twist in the String-Tale

After achieving such a massive feat, I now shrink the task and hand a perfectly spherical basketball over to you. This basketball is 24 cm in circumference around its equator. You are now to repeat the same drill.

Firstly, you will be wrapping the basketball tightly with a piece of string (around its equator). Since the basketball is 24 cm in circumference, this string will also be 24 cm long. Next, you will be wrapping the basketball with a second-string and this string needs to hover at a height of 1 cm above the basketball’s surface.

Question: How much longer does the second string need to be compared to the first string in order to hover 1 cm above the basketball’s surface? Again, it is sufficient if you have a vague idea about the answer. Once you have an idea, hold this at the back of your mind as well. Now, let’s have a look at the answers!

The Answers

In the case of the Earth, the second string needs to be approximately 6.28 cm longer than the first string to enable it to hover 1 cm above the earth’s surface.

Are you surprised that it is such a low number? Or did you get the answer (or its rough scale) right? Before you veer off to the comment’s section, let us have a look at the rest of the puzzle as well.

In the case of the basketball, the second string needs to be approximately 6.28 cm longer than the first string to enable it to hover 1 cm above the basketball’s surface.

That’s interesting. We have the same number here. In fact, both strings need to longer by the same number of centimetres to hover 1 cm above the respective surfaces. Did you see this coming? Or did the answers surprise you?

The Anchoring Effect

Most people guess that the answer to the puzzle in the case of the earth has to be a very large number, whereas, in the case of the basketball, it has to be a much smaller number. This human behaviour is explained by the anchoring effect.

It is a cognitive bias that occurs when our decisions are influenced by particular reference points or numbers we hear around the time or context of the decision. In the case of the earth, we are dealing with a huge number — 40,000 Km. Therefore, we are more likely to guess a big number as the answer. In the case of the basketball, we are dealing with a comparatively smaller number — 24 cm. As a result, we are likely to guess a much smaller number as the answer.

All of this typically happens without our conscious awareness; hence the term cognitive bias. Now that you have an understanding of how this puzzle affects human behaviour, let us go ahead and unravel the logic behind this puzzle.

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The Logic Behind the Puzzle

The trick here is to realise that we are dealing with circumferences. The mathematical formula for circumference is 2πR, where R is the radius.

When the second-string hovers 1 cm above the Earth’s surface, its radius increases 1 cm compared to the first string. The length of the second-string is given by its circumference. The difference between the circumferences of both strings would give us the answer of how much longer the second string needs to be. This can be calculated as follows:

[Circumference of Second String] — [Circumference of First String]

= [2*π*R-Second-String] — [2*π*R-First-String]

= 2*π*[(R-Second String) — (R-First String)]

= 2*π*1 (since the difference in radii is 1 cm)

~ 6.28 cm (approximate answer)

As to why we get the same answer for the basketball as well, it helps to realise that the answer does not depend on the actual scale of the radius of the earth or the basketball, but only on the difference in radii between both the strings. The difference in radii between the strings in the case of the basketball is also 1 cm. Hence, we end up with the same result.

Final Thoughts

Even though this turns out to be a simple puzzle, I feel that it imparts a valuable lesson. Whenever we are dealing with important numerical/financial decisions, it makes sense to stop listening to the so-called ‘gut-feeling’ and just work out the answers.

It feels painful at that particular moment, but it leads to a much clearer view of the problem at hand.

We might not benefit from wrapping strings around the Earth. But wrapping our heads around real-life problems is always a welcome advantage!

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Further reading that might interest you: How To Use Science To Win At Rock-Paper-Scissors? and What Really Happens When You Divide By Zero?