Slack

Wrap a rope tight around the Earth's equator, then splice in enough extra rope to lift the whole loop exactly 1 meter off the ground, uniformly, all the way around — not touching anywhere. How much longer does the rope have to be? Most people's first guess runs to kilometers, because the Earth is enormous. The real answer is about 6.28 meters. Not 6.28 kilometers — meters. A version of this puzzle appears in an English clergyman and mathematician's 1702 student's book on Euclid, and it still catches people out for the same reason it did then: intuition wants the answer to depend on how big the circle already is, and it doesn't, at all.

The reason is one line of algebra, not a trick. A circle's circumference is C = 2πr — radius in, circumference out, nothing else. Raise the rope by a height h and its new radius is r + h, so the new circumference is 2π(r + h), which is 2πr + 2πh. The first term is just the old circumference again; the only new length is 2πh. The original radius, r, cancels out completely — it never makes it into the answer. Every extra meter of radius buys exactly 2π meters of new circumference, whether that meter of radius is being added to a basketball or a planet.

Watch the numbers beside the picture rather than the picture itself — the picture can't be to scale (nothing could draw a basketball and the Sun on one canvas honestly), but those numbers are the real ones, computed live for whichever body is selected, both circumferences shown in plain meters so the digits that move are directly comparable. Switch bodies without touching the slider and the extra rope needed line never moves. Everything else about the body — both circumferences, how many digits they run to — changes by orders of magnitude. That one line doesn't, because it was never a function of the body to begin with.

Push the slider further and the basketball case gets stranger still: raise its rope 1 meter and the loop's new radius is roughly ten times the ball's own radius — you'd be looking at a hoop nearly a meter wide floating around a ball you can hold in one hand. It still only took the same 2π meters of extra rope Earth's loop took, because the formula never asked how big the ball was either.

Honest gap: every body here is stood in for a perfect circle in cross-section, which none of them actually are — Earth bulges at the equator, a basketball has seams and panels, and the Sun has no solid surface at all, only a photosphere the radius below is defined against. None of that matters to the result, which is the point being demonstrated, not a shortcut being taken: whichever radius is used for a given body, the extra rope a given height needs is 2πh regardless. The two circles drawn are a fixed, identical size for every body — arbitrary, and not to scale of one another, since no single picture could hold a basketball and the Sun at the same scale. The gap between those two circles is not exaggerated, though: it is drawn at one fixed real scale, pixels per meter of height, the same scale for every body — which is why the gap looks identical no matter which body is selected. It looks identical because it is.

Sources: David Darling's rope-around-the-Earth puzzle entry, for the 1702 William Whiston attribution (a students' book on Euclid) and the puzzle's classic framing — a fixed extra length of rope, run the other way to solve for the gap it lifts, rather than this page's fixed height solved for the rope. NASA's Earth Fact Sheet for the WGS84 equatorial radius, 6,378.137 km. NASA's Moon Fact Sheet for its volumetric mean radius, 1,737.4 km. The IAU's 2015 Resolution B3 for the nominal solar radius, 695,700 km. Wikipedia's basketball (ball) entry for the NBA/FIBA official size-7 circumference, 29.5 in (749.3 mm), which this page's own basketball radius is computed straight from — not looked up separately. The 2πh derivation is this page's own algebra, not any of theirs.