Overview
One extra metre on a forty-thousand-kilometre rope sounds like nothing at all. It lifts the rope far enough to crawl under.
How to solve Rope Around the Earth
- Write the circumference as C = 2πR, so the radius is R = C/2π.
- The new radius is (C + 1)/2π = R + 1/2π. Subtract: the gap is exactly 1/2π.
- 1/(2π) ≈ 0.159 m ≈ 16 cm, and R has vanished from the answer.
The key insight
The relationship between circumference and radius is linear, so the change in radius depends only on the change in length: ΔR = ΔC/2π. Our intuition keeps comparing 1 metre against 40 000 km, but that ratio never enters the calculation.
Variations & echoes
- Do it to a tennis ball and the gap is still 16 cm — proportionally enormous, absolutely identical.
- The harder cousin: lift the extra metre at a single point instead of uniformly, and the peak height jumps to about 122 metres.
Frequently asked questions
Is the answer really independent of the planet?
Yes. ΔR = ΔC/2π contains no R at all, so the same extra metre lifts any circle by the same 16 cm.