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.
Tips
- Radius is half the diameter – double-check which one you measured.
- Keep π exact until the end, or use at least 3.14159.
- Arc length and sector area need the central angle in radians in the formulas.
- Type any result to work backwards to the radius.
Common mistakes to avoid
- Using the diameter in a formula that needs the radius.
- Forgetting to square the radius when finding the area.
- Using degrees in s = rθ, which needs radians.
Inputs explained
| Field | What to enter |
|---|---|
| Sphere | Choose one: Earth (R = 6,371 km), Moon (R = 1,737.4 km), Basketball (R = 12 cm), Custom radius |
| Sphere radius | Units: cm, m, mm, in, ft, yd … |
| Added string length ΔL | Units: 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) |
|---|---|
| 50 | 15.9155 |
| 75 | 15.9155 |
| 100 | 15.9155 |
| 125 | 15.9155 |
| 150 | 15.9155 |
| 200 | 15.9155 |
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)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.
Frequently asked questions
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.
What formula does the String Girdling Earth Calculator use?
Even gap: h = ΔL / (2π) (independent of the sphere size).