String Girdling Earth Calculator (add rope to a belt round the Earth: how high?)

Understanding the rope around the Earth calculator — concept and principle

This calculator solves the classic "rope around the Earth" puzzle. A rope fits snugly around a sphere; you add a little extra length. How high does the rope rise if it is lifted evenly all round, and how high if it is pulled up at a single point? Use it for maths puzzles, teaching intuition about circles and tangents, and surprising a classroom.

Spread evenly, 1 m of slack lifts the rope only 1 ÷ 2π m ≈ 15.9 cm; pulled up at one point it reaches about 121 m.

How it is calculated

Lifted evenly: 2π(R + h) = 2πR + ΔL, so h = ΔL ÷ 2π. The radius R cancels.
Pulled up at one point: with θ the half-angle of the lifted arc, the rope follows two tangent lines, so ΔL = 2R(tan θ − θ), and the peak height is H = R(1 ÷ cos θ − 1).
The first equation is solved for θ by bisection, using the series tan θ − θ = θ³/3 + 2θ⁵/15 + … for tiny angles to avoid rounding loss.
The page also gives the arc of rope off the ground (2Rθ), each straight tangent piece (R·tan θ), the original rope length 2πR and the full angle of the lifted arc (2θ).

The principle

The circumference grows by 2π for every unit of radius, whatever the size of the circle. So one extra metre of rope always gives the same gap, about 15.9 cm, for a football or a planet. That feels wrong because 1 m is tiny compared with 40,000 km, but the gap depends only on the added length. When the slack is gathered at one point, the rope leaves the surface along two tangents. On a huge sphere, the tangents run a long way before they meet, and a very small extra length can then lift the peak surprisingly high.

Worked example

Default inputs: R = 6,371 km (Earth's mean radius), ΔL = 1 m.

Even lift: h = 1 ÷ 2π ≈ 0.1592 m = 15.92 cm.
Original rope: 2π × 6,371 km ≈ 40,030.174 km.
Pulled at one point: solving 2R(tan θ − θ) = 1 m gives θ ≈ 0.354°, so the lifted arc spans 0.7076°.
Peak height H = 6,371,000 × (1 ÷ cos θ − 1) ≈ 121.46 m.
Rope off the ground ≈ 78.68 km of surface; each straight piece ≈ 39.34 km.

Each point is the calculator re-run with only “Extra rope length ΔL (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Extra rope length ΔL (m)	Gap if lifted evenly all round (cm)
0.25	3.98
0.5	7.96
0.75	11.94
1	15.92
1.25	19.89
1.5	23.87
2	31.83

How to read the result

The even-lift gap is the same for any sphere and grows in direct proportion to the extra length. The one-point height grows roughly with the cube root of R²ΔL, so it is much larger on big spheres. Both answers assume a perfectly smooth sphere and a rope that does not stretch.

Common mistakes

Expecting the even gap to depend on the size of the sphere.
Entering the radius in metres when the field asks for kilometres.
Using the diameter instead of the radius.
Thinking the one-point height scales linearly with the extra rope; it grows much more slowly.

Keep in mind: This is an idealised geometry puzzle. Real terrain, rope stretch and sag under its own weight are ignored.

Frequently asked questions

Why is the gap the same for the Earth and a football?

Because circumference = 2π × radius, so adding ΔL to the circumference always adds ΔL ÷ 2π to the radius.

How high is the gap with 1 m extra?

About 15.9 cm all round, enough for a cat to crawl under.

Why is the pulled-up version so high?

The rope is nearly tangent over tens of kilometres, and a tiny extra length lets those long tangents meet well above the ground.

Sources: Standard geometry: circumference 2πr; tangent length R·tan θ and arc length Rθ

Checked 9 Nov 2026. Standard result for the “rope around the Earth” puzzle (circumference 2πr, so the gap is ΔL ÷ 2π); the pulled-up variant follows from the tangent length R·tan θ from an external point and the arc length Rθ (standard geometry, derived here). Earth’s mean radius 6,371 km (standard value).

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