The Rope Around the Earth

Imagine a rope wrapped snugly around the Earth's equator, then imagine adding just one extra meter of rope and pulling it into a perfect circle, lifted evenly off the ground all the way around. How big is the gap between the ground and the rope? Most people guess it must be microscopically small, since one meter is nothing next to the Earth's roughly 40,000-kilometer circumference. The actual gap is about 16 centimeters — big enough to slide a large book underneath, everywhere around the entire planet.

The trick is that the answer doesn't depend on the circle's size at all. Circumference is $C = 2\pi r$, so adding a fixed length $\Delta C$ to the rope changes the radius by $\Delta r = \Delta C / (2\pi)$ — a formula with no $r$ in it. Whether the "planet" is a basketball or the Earth, adding one extra meter of rope always lifts it by the same $1/(2\pi) \approx 0.159$ meters, about 16 cm.

Worked example. Adding exactly 1 meter of rope: $\Delta r = 1 / (2\pi) \approx 0.159$ meters, or about 15.9 cm, regardless of whether the sphere is the Earth or a soccer ball. Adding 2 meters instead simply doubles the gap to roughly 31.8 cm — the relationship is exactly linear in the extra length, never in the planet's size.