String Girdling Earth Calculator

See how far a uniformly raised string moves from a sphere when a fixed length is added to its circumference.

Calculate the string gap

Use consistent units for the sphere radius and the added string length.

Earth radius

Extra string length

About the string girdling Earth problem

The string girdling Earth puzzle asks what happens when a taut string circles a sphere and a small fixed length is added. If the longer string is lifted so that it remains a perfect concentric circle, how large is the uniform gap between the string and the surface? The surprising answer does not depend on the original size of the sphere. It is determined only by the added length. A circle with radius r has circumference C = 2 pi r. After adding length L, its new circumference is C + L and its new radius is (C + L) divided by 2 pi. Subtracting the original radius cancels C and leaves the gap L divided by 2 pi. Adding one meter therefore raises the string by about 0.159155 meter, or almost 15.9 centimeters, whether the original circle surrounds Earth, a basketball, or a coin. The radius field is included so the calculator can also report the original circumference and the new radius. It does not affect the computed gap. This cancellation is the central lesson of the puzzle: circumference changes at a constant rate of 2 pi units for each unit of radius. A very large circle does not require a larger added length than a small circle to create the same radial increase, provided the string remains circular and evenly spaced. The idealized model makes several assumptions. Earth is treated as a perfect sphere, the string has no thickness or elasticity, and the raised loop remains centered. A real rope would sag under gravity, encounter terrain, stretch, and require supports. Those practical effects make a uniform physical gap impossible without additional structure. The calculator addresses the mathematical thought experiment rather than an engineering installation. Use the same unit for both entries. If the radius is in meters and the extra length is in meters, all reported lengths are meters. You can use kilometers, feet, inches, or any other consistent unit. This problem is a memorable demonstration of algebraic cancellation, proportional reasoning, and the linear relationship between a circle's radius and circumference. It also helps build intuition about why changes in circumference can behave differently from changes in area or volume.

String girdling examples

The same added length always produces the same gap.

Sphere and added length	Uniform gap	Observation
Earth radius 6,371,000 m; add 1 m	0.159155 m	About 15.9 centimeters
Radius 100 m; add 10 m	1.591549 m	Radius does not change the gap formula
Radius 1 ft; add 6.283185 ft	1 ft	Adding 2 pi raises the loop by one unit

How to calculate the raised string gap

Enter the radius of Earth or any spherical object.
Enter the amount of length added to the original tight string.
Select Calculate Gap to compute the original circumference and new radius.
Read the uniform gap, which equals the extra length divided by 2 pi.

String girdling Earth FAQ

Why does the Earth's radius not affect the gap?

The original circumference cancels when the old radius is subtracted from the new radius. Only the added length divided by 2 pi remains.

What gap does one extra meter create?

One meter divided by 2 pi is approximately 0.159155 meter. That is about 15.9 centimeters at every point in the idealized model.

Does the result work for a small ball?

Yes, the uniform radial increase is identical for any starting radius. The loop must remain circular and centered for the relationship to apply.

Can I mix feet and inches?

Convert both inputs to one common unit before calculating. The output then uses that same unit and can be converted afterward if needed.

Would a real rope float uniformly above Earth?

No, gravity, terrain, elasticity, and supports would change the physical shape. This calculator models a perfect concentric circular string as a mathematical thought experiment.