# String Girdling Earth Calculator

URL: https://speedycalculator.com/calculators/math/string-girdling/
Last updated: 2026-10-30

> The string girdling Earth calculator answers the classic puzzle: if a string tight around a sphere is lengthened by ΔL, how high does it float evenly – and how high can it be pulled up at a single point?

## Inputs

- Sphere
- Sphere radius (cm, m, mm, in, ft, yd, km, mi)
- Added string length ΔL (cm, m, mm, in, ft, yd, km, mi)

## Results

- Even gap all round
- Height when pulled up at one point
- Original circumference

## Formula

```
Even gap: h = ΔL / (2π)  (independent of the sphere size)
Pulled at one point: tan θ − θ = ΔL/(2R),  h = R(1/cos θ − 1)
```

## Explanation

Add just 1 m to a string around the Earth's equator and it rises about 16 cm all round – the same as for a string around a football. Yet pulled up at one point, the same metre lifts it over 120 m. Both answers surprise almost everyone.

### How to use the String Girdling Earth Calculator

- **Sphere**
- **Sphere radius**
- **Added string length ΔL**

Results update as you type, and everything is calculated in your browser.

### How it works

Choose the sphere and the added length. The even gap uses ΔL/2π. The single-point lift solves tan θ − θ = ΔL/(2R) numerically, where θ is half the angle of the string's free part.

### Inputs explained

FieldWhat to enterSphereChoose one: Earth (R = 6,371 km), Moon (R = 1,737.4 km), Basketball (R = 12 cm), Custom radiusSphere radiusUnits: cm, m, mm, in, ft, yd …Added string length ΔLUnits: cm, m, mm, in, ft, yd …

### What each result means

- **Even gap all round** = [Added string length ΔL]÷(2×pi)
- **Height when pulled up at one point** = [Radius]×(1÷cos([θ7]) - 1)
- **Original circumference** = 2×pi×[Radius]

### Example values

How the main result changes when **Sphere radius** changes and everything else stays at the example values above:

Sphere radius (cm)Even gap all round (cm)5015.91557515.915510015.915512515.915515015.915520015.9155

## Worked example

Inputs — Sphere: Earth (R = 6,371 km); Added string length ΔL: 100 cm.

Results — **Even gap all round = 15.915494 cm**; Height when pulled up at one point = 12,146.072318 cm; Original circumference = 4,003,017,359.204114 cm.

## Key terms

- **Tangent**: A line touching a circle at one point.

## FAQ

**Why does the gap not depend on the size of the sphere?**
C = 2πr, so ΔC = 2πΔr – the change is linear.

**How much string to raise it 1 m around the Earth?**
2π ≈ 6.28 m.

**Why is the single-point lift so large?**
The extra length is spread over a long tangent section near a huge sphere.

**Is the even-gap answer exact?**
Yes, for a perfect sphere.


## Sources

- [Wolfram MathWorld – Circle](https://mathworld.wolfram.com/Circle.html)