How the String Girdling Earth Calculator works
"String girdling the Earth" is a classic recreational-math puzzle: imagine a string wrapped snugly around the Earth's equator, forming a perfect circle. Now splice in a little extra string and let the loop expand into a slightly larger, concentric circle that floats a uniform height above the ground everywhere. How much does that gap grow for a given amount of extra string? The surprising answer is that it does not depend on the size of the sphere at all — the same extra length lifts a string off a basketball or off the entire Earth by exactly the same amount.
Formula and derivation
A circle's circumference is C = 2πr, where r is the radius. If you add an extra length ΔL to the string, the new loop is still a circle, now with radius r + Δr, so its circumference is C + ΔL = 2π(r + Δr) = 2πr + 2πΔr. Subtracting C = 2πr from both sides leaves ΔL = 2πΔr, so solving for the gap gives Δr = ΔL / (2π). Notice that r cancelled out completely — the radius increase depends only on how much extra string you add, never on the original size of the sphere. This calculator converts your radius and extra-length inputs to a common unit, applies Δr = ΔL / (2π) to get the uniform gap, then reports the new radius and both circumferences.
Why this is counterintuitive (and correct)
Intuition says wrapping something as large as the Earth should require an enormous amount of extra rope to create any noticeable gap, while a basketball should need almost none. The math says otherwise: because both circles are related to their radii by the same factor of 2π, the r terms always cancel. Adding just 1 meter of extra rope to a string girdling the Earth's roughly 40,075 km equator raises it about 15.9 cm off the ground everywhere — the exact same lift you'd get by adding 1 meter of string around a basketball. The puzzle is a favorite example in math education precisely because it shows how a small, constant addition produces a size-independent, uniform effect.