The Elastic Band Paradox: A Meter That Lifts the Band 16 Centimeters

Imagine a ring that fits perfectly on your finger. Cut it and add a meter of material. It's easy to imagine that if you put it back on your hand, the ring will be very loose, since its new diameter is clearly much larger. Now imagine an elastic band that fits snugly around the Earth, encircling it at the equator, and you stretch its perimeter by a meter. The band will now be slightly larger than before, but how far does it separate from the planet's surface? The answer, as always, defies common sense.

Although the elastic band paradox is not considered a paradox in the strict sense, it does clash with our common sense, because it has a seemingly impossible solution. The problem is stated as follows: imagine a sphere the size of planet Earth is surrounded by an elastic band that fits perfectly around its equator.

That band measures approximately 40,000,000 meters in length. If the band is stretched by a meter, to a length of 40,000,001 meters, how far does it separate from the sphere's surface? Could we slide a piece of paper, a coin, or a tennis ball between the band and the sphere?

A Smaller-Scale Experiment

Before we start calculating and trying to determine the separation, let's use common sense to solve a similar situation, but on a much smaller scale. Imagine, as we said at the beginning, a ring that fits perfectly on one of your fingers.

Suppose your finger has a radius of 1 centimeter, so the ring's circumference is about 6.28 centimeters (2 × π × r). If we cut the ring and add a meter of material, when we put it back together its circumference is now much larger, and it has practically become a hula hoop that your whole body could pass through. Placing it on a finger again, you would see a gap between the finger and the new ring of about 16 centimeters.

The previous experiment seems to make sense. After all, we've turned a ring with a 6.28-centimeter circumference into one of more than 106 cm, so we're not surprised it's so far from the finger. The distance from the center of the finger can be calculated using R = circumference / 2 / π, giving roughly 16 centimeters. This is one of those reassuring cases where common sense and reality go hand in hand.

The Earth-Sized Band

The band around the sphere has a length (or perimeter) of 40 million meters. Its radius, according to the formula above, is 40,000,000 / 2 / π = 6,366,197.83 meters, about 6,366 kilometers. Common sense tells us—more like shouts at us—that if we stretch such a huge perimeter by the same single meter we added to the ring, the separation from the enormous sphere will be imperceptible.

Perhaps the reason is that the percentage increase in the first case is close to 1600%, while here it's about 0.000000025%. With such a tiny increase, it's hard to imagine being able to slide a sheet of paper between the sphere and the elastic band. Or is it?

The Surprising Result

But, once again, we're facing one of those cases where common sense plays a trick on us. Let's do the math: the radius of the circle formed by the stretched band can be calculated as R = 40,000,001 / 2 / π = 6,366,197.99 meters. The result looks very similar to the radius of the unstretched band (6,366,197.83). But if we look closely, we see that that single meter we added is enough to separate the band from the sphere by 6,366,197.99 – 6,366,197.83 = 0.16 meters. The same 16 centimeters as in the finger and ring case! Common sense just surrendered, again, to reality.

As you can see, the result is independent of the radius of the sphere or the finger. It only depends on the length we add and the value of π. Since we added one meter to the ring and stretched the elastic band by one meter, both separate from their original surfaces by the same 16 centimeters.

Returning to the original question, we could easily slide a tennis ball between the stretched band and the sphere. We've done the calculations using only a couple of decimals, but that doesn't change the result much. The actual value is around 15.9154943... centimeters. Regardless of how many decimal places you consider, the result will be the same in both cases. What do you think?