The length of the equator
The Earth's equator is about 40,075 km long. Because the Earth spins it is slightly flattened, so a circle through the poles is shorter, about 40,008 km.
A rope is tied tightly around the equator, then made 1 metre longer and lifted evenly off the ground. Most people expect a gap thinner than a hair, because a metre is lost next to 40 thousand kilometres. In fact a mouse could walk under the rope. And the answer is the same for an orange, the Earth or the Sun.
This is an old geometry riddle that often turns up in puzzle books. It tests how we handle very large and very small numbers when they meet in one problem.
A rope is tied tightly around the Earth along the equator. Its length equals the equator, about 40,075 km. We treat the Earth as a perfect sphere with no mountains or oceans.
The rope is cut and a 1 metre piece is spliced in. Now it is one metre longer than the equator.
The rope is lifted off the surface so that it is at the same height everywhere. It forms a circle again, just a slightly larger one.
How big is the gap between the rope and the ground? Could an ant get under it? A cat? A person?
"One metre over 40 thousand kilometres? The gap will be microscopic"
In fact the gap is about 15.9 cm. A mouse could walk under the rope without ducking. The mistake is comparing the metre with the whole equator, when all that matters is how a change in length relates to a change in radius.
The real equator has mountains and valleys, but the puzzle is about geometry, not terrain. Another way to lift the rope is in round 5.
Move the slider to your answer. The scale is logarithmic, from a micrometre to 10 metres. Below the slider you can see who would fit under the rope if you were right.
Your answer will be compared with the right one here.
Pick a sphere and how much rope to add. On the left is the whole sphere; on the right is a zoomed window at the top, true to scale, with a cat for comparison. For the Earth the rope merges with the surface in the full view, but in the zoom the gap is the same for every sphere.
For the Earth the relative lengthening is tiny, but the Earth's radius is huge. A tiny fraction of a huge number comes out as an ordinary 16 cm.
The computer builds a rope out of 100,000 straight pieces around a sphere of radius R, adds a piece, and finds the new radius by bisection, without using 2πR. The dots show the result for radii from a centimetre to a million kilometres. For comparison, the chart also shows the height of the "tent" from round 5, which does depend on the size of the sphere.
| Sphere | Radius | Rope length | Relative lengthening | Gap | Tent |
|---|
A circle's length is proportional to its radius. So a change in length is proportional to a change in radius, and the size of the sphere cancels out. Here are five ways to see it.
Each metre of radius adds 2π ≈ 6.28 m to the circle. And the other way round, each metre of circle adds 1/(2π) ≈ 0.159 m to the radius. The rule is the same for circles of any size. Move the slider: R changes by a factor of billions, and the gap doesn't.
Intuition is right about one thing: the rope grew by a tiny proportion. But the radius grows by the same proportion, and the Earth's radius is 6,378 km. A tiny fraction of thousands of kilometres is centimetres. The mistake is carrying the "microscopic" proportion over to the gap itself.
A standard running track has two straights and two semicircular bends. The straights are the same for every lane, and the two bends together make a full circle. So each lane out is about 2π × 1.22 ≈ 7.67 m longer per lap, however big the arena. Without a staggered start, lane 8 would be about 54 m longer than lane 1.
The rules measure distance slightly inside each lane, so real start staggers differ a little from this simple estimate.
With an even lift R cancels, so the gap does not depend on the size of the sphere. In the tent R stays in the formula: the height grows roughly as the cube root of R. So the tent over the Sun is taller than over the Earth, and the one over an orange is lower. For small angles tan θ − θ ≈ θ³/3, which gives the approximation.
Now the rope is not lifted evenly. It lies on the ground, and at one point it is pulled upward until it is taut. The result is a tent: two straight lines tangent to the sphere, meeting at the peak. How tall is it for the Earth and one extra metre?
A circle and a square with the same perimeter as the equator. The rope gets the same extra length as above and is lifted in different ways.
| Shape | How we lift | Height |
|---|
The Earth's equator is about 40,075 km long. Because the Earth spins it is slightly flattened, so a circle through the poles is shorter, about 40,008 km.
In late 18th-century France the metre was defined as one ten-millionth of the distance from the North Pole to the equator. That is why a circle around the Earth through the poles comes out at about 40,000 km.
To lift a rope around the Earth evenly by 1 metre, you only need to add 2π ≈ 6.28 m. For a person 1.8 m tall to walk under it, about 11.3 m is enough.
A lane on a stadium track is 1.22 m wide. Over the full circle of the bends, each lane out is about 2π × 1.22 ≈ 7.67 m longer. That is why races with bends use staggered starts.
If the extra metre of rope is pulled up at a single point, the peak of the tent rises to 121.5 m. The rope leaves the ground about 39 km from the peak on each side.
The Sun's circumference is about 4.37 million kilometres, 109 times the Earth's equator. Yet one extra metre of rope still gives a 15.9 cm gap there, just as for an orange.
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