String Girdling Earth Calculator

The String Girdling Earth Puzzle

Imagine a string tightly wrapped around Earth's equator. If you raise it uniformly 1 meter above the ground, how much longer must the string be? Surprisingly, only about 6.28 meters ($2\pi$ m), regardless of Earth's size. This counterintuitive result is the string girdling Earth (rope around Earth) problem.

See also the Circle Calculator and Circle Equation Calculator.

The Key Formula

For a gap height $d$ and extra string length $\Delta L$:

$$\Delta L = 2\pi d$$

Earth's radius cancels out, so this works for any sphere: Earth, Mars, a basketball, or a coin.

Worked Example

If you splice 1 meter of string into the loop, the gap is:

$$d = \frac{\Delta L}{2\pi} = \frac{1}{2\pi} \approx 0.159 \text{ m} \approx 16 \text{ cm}$$

That is enough room for a cat to pass underneath.