How this calculator works
Imagine a string pulled tight around a sphere and then lengthened by a small amount. When the longer loop is pulled tight again, it floats uniformly above the surface. The surprising part is that the height does not depend on the size of the sphere. Adding one metre of string lifts the loop by one metre divided by 2pi, which is about 15.9 centimetres, whether the sphere is a basketball or the Earth. The reason is that both circumferences grow by the same added length, and the extra radius is the extra circumference divided by 2pi. This page solves in both directions: give it the added string and it reports the lift, or give it the desired lift and it reports the extra string needed.
Formula
Added length L and lift h are related by L = 2πh, so h = L/(2π). The original circumference is C = 2πR and the new circumference is C + L. The new radius is R + h. The lift does not contain R, which is why the sphere size cancels out.
Worked example
Adding 1 m of string around Earth, whose mean radius is about 6371 km, lifts the loop by 1/(2π) ≈ 0.1592 m, about 16 cm. The original circumference is about 40,030 km and the new one is 1 m longer. To lift the string by 1 m instead, you need 2π ≈ 6.283 m of extra string.
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Frequently asked questions
Why does the size of the Earth not matter?
The lift depends only on how much longer the new loop is than the old one. Both loops are circles, and circumference grows linearly with radius. Adding a fixed length adds a fixed amount to the radius, no matter where the loop started. The starting circumference cancels in the subtraction.
How much string lets a person walk under it?
A person needs roughly 2 m of clearance, which takes about 4π ≈ 12.57 m of extra string. The same amount works around Earth or around a marble, because only the lift matters.
Is this physically possible?
The calculation assumes a perfectly flexible string pulled tight and a perfectly smooth sphere. Real string stretches, real terrain is uneven and Earth is not a perfect sphere, so the practical result would be changed by those effects. The mathematical result is exact for the ideal model.
What if I add no length?
Zero added length gives zero lift and the new circumference and radius equal the originals. Zero is allowed, and the calculator returns the unchanged circle.