String Girdling Calculator — Rope Around the Earth Puzzle
The string-girdling puzzle asks: if you lay a rope tightly around the Earth and then add just 1 metre of extra rope, how high above the surface does the rope rise? The answer — about 15.9 cm — is the same for ANY sphere, regardless of size. This calculator shows why.
m
km
Δr = ΔC ÷ (2π) — this height is the same for ANY original circumference
Original circumference–radius relationship
After adding ΔC, new circumference gives new radius r₁
Subtracting the two equations
Solving for height gained — no C₀ term remains
- 1
Divide extra rope by 2π
1 m ÷ 6.283185 = 0.159155The original circumference cancels — Δr depends only on the extra rope. - 2
Convert Δr to centimetres
0.159155 m × 100 = 15.9155
How does this calculator work?
Adding ΔC metres of rope to a loop around any sphere lifts it Δr = ΔC / (2π) ≈ ΔC × 0.159 metres above the surface — independent of the sphere's size. For 1 m of extra rope the rope rises ~15.9 cm whether the sphere is a tennis ball or the Earth. The original circumference cancels out of the algebra.
Formula
How this is calculated
For any sphere of circumference C₀ the radius is r₀ = C₀ / (2π). If you add ΔC metres of extra rope and let it hover uniformly above the surface at height Δr, the new circumference is C₀ + ΔC = 2π(r₀ + Δr). Subtracting the first equation from the second gives ΔC = 2π Δr, so Δr = ΔC / (2π). The original circumference C₀ — and therefore the size of the sphere — cancels out entirely.
For ΔC = 1 m: Δr = 1 / (2π) ≈ 0.1592 m = 15.92 cm. This is the same whether the sphere is a tennis ball (circumference ≈ 21 cm), the Earth (≈ 40 075 km), the Sun (≈ 4.4 million km), or any other size. The result is so counterintuitive because humans have good intuition for linear changes but not for the relationship between circumference and radius, where a 1:2π conversion factor applies.
The puzzle has a second part: how much extra rope do you need to lift the rope exactly 1 metre above the Earth? Answer: ΔC = 2π × 1 m ≈ 6.28 m — just over 6 metres of extra rope, added to a rope already 40 075 km long. Equally surprising in the other direction.
Frequently asked questions
Because ΔC = 2π × Δr. The original radius r₀ and the original circumference C₀ cancel out when you subtract the old and new circumference equations. The only variables left are ΔC (extra rope) and Δr (height gained), connected by the constant 2π.
Rearranging: ΔC = 2π × Δr = 2π × 1 m ≈ 6.283 m. So adding only about 6.3 metres to a rope that is already 40 075 km long would raise the entire loop uniformly 1 metre above the surface — another striking illustration of the same principle.
Yes — the result Δr = ΔC / (2π) holds for any closed curve, not just circles. For a convex closed curve, the Cauchy–Crofton formula shows that adding ΔC to the perimeter increases the width (in all directions) by ΔC / (2π), independent of the original shape.
Also known as
TG we-Calculate Editorial Team. (2026). String Girdling Calculator — Rope Around the Earth Puzzle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/string-girdling-calculator
TG we-Calculate Editorial Team. "String Girdling Calculator — Rope Around the Earth Puzzle." TG we-Calculate. 2026. https://we-calculate.com/calculator/string-girdling-calculator.
TG we-Calculate Editorial Team, "String Girdling Calculator — Rope Around the Earth Puzzle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/string-girdling-calculator
@misc{wecalculate_string_girdling_calculator, title = {String Girdling Calculator — Rope Around the Earth Puzzle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/string-girdling-calculator}}, year = {2026}, note = {TG we-Calculate} }
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