The most-repeated version of the fact gets both adjectives backwards. The Earth is round enough to be a regulation ball — and nowhere near smooth enough. The error is one category mistake, and it is recomputed here from the published numbers.
You have heard it. It is one of those facts that travels well, printed in magazines and recited in classrooms:
"If you shrank the Earth down to the size of a billiard ball, it would actually be smoother than a real billiard ball … the Earth is smooth enough, but not round enough, to be a regulation ball." — the popular form, after Phil Plait, "Ten Things You Don't Know About the Earth," Bad Astronomy / Discover, 2008
It is a lovely fact, and it is wrong in a precise and instructive way. Not off by a little — inverted. Checked against the actual geodetic constants and the actual equipment rules, the Earth is round enough to pass as a regulation billiard ball, and it is not remotely smooth enough. The slogan swaps the two verdicts, and it does so for a single reason: the only number a billiard ball's rulebook publishes is a tolerance on its diameter. That is a roundness spec. The slogan reads it as a smoothness spec.
So let us hold the ball up to the light. Every figure below is recomputed in your browser from the constants in the table at the foot of the page; nothing here is asserted that the page does not also compute.
A regulation pool ball, by the World Pool-Billiard Association equipment spec, is 2¼ inches across — 57.15 mm — with a published tolerance of ±0.005 in (±0.127 mm) on that diameter. That is the whole of the relevant number. There is no separate sphericity figure, and no surface-roughness figure, in the rules at all. Keep that in view: it is the hinge of the entire confusion.
And the Earth is not a sphere. It spins, so it bulges at the equator and flattens at the poles — the WGS84 reference ellipsoid (the datum your phone's GPS uses) has an equatorial radius of 6,378,137 m and a polar radius 21,385 m shorter. So the Earth's diameter through the equator is about 42.8 km longer than through the poles. That bulge is the single biggest way the planet departs from a ball. Its surface bumps — the mountains and trenches — are, by comparison, almost nothing: top of Everest to bottom of the Challenger Deep is a relief of only 19.8 km.
Two different questions, then, and two different Earth-numbers to answer them with. Is the Earth round enough? — compare its 42.8 km bulge to the ball's diameter tolerance. Is the Earth smooth enough? — compare its 19.8 km of relief to the ball's surface finish. Shrink everything to 57.15 mm and look.
Below is one cue ball, scaled from the real Earth by diameter. The two departures from a perfect sphere — the equatorial bulge and the surface relief — are far too small to see at true size (that is exactly why the fact feels true). So drag the magnifier to swell them until the eye can judge what the numbers already know. The verdict underneath is driven by the figures, never by the picture.
At true scale the bulge and the gouge both disappear into the line of the circle — your eye cannot referee this contest. The numbers can, and they split the two questions cleanly apart.
The ball's diameter may sit anywhere in a band 0.010 in wide (from −0.005 to +0.005) — that is the most a regulation ball's diameter is allowed to vary, which is the closest the rules come to bounding its roundness. Scaled up to Earth's size, that band permits a diameter difference of about 56.7 km. The Earth's actual equator-to-pole diameter difference is 42.8 km. It fits, with room to spare: the Earth is round enough to be a regulation billiard ball.
This is where the popular hedge — "not round enough" — goes wrong. It tends to compare the bulge against half the band (the bare ±0.005), or to mismatch the percentages, and concludes the bulge busts the tolerance. Read the tolerance as what it is — the full width a diameter may occupy — and the bulge sits inside it. The roundness verdict is genuinely sensitive to that reading, because the WPA never wrote a sphericity spec; that ambiguity is part of the honest answer, and the instrument lets you flip between the two readings.
Now the other adjective. Surface smoothness is not about how much the diameter varies; it is about how far the surface strays from the ideal at any point — the finish. The rulebook says nothing about finish, so the slogan quietly borrows the diameter tolerance to stand in for it: Earth's relief, shrunk to the ball, is a bump 0.089 mm tall, which is smaller than the 0.127 mm tolerance, so — "smoother." But that compares a bump on the surface to a tolerance on the size. They are different kinds of thing. It is like certifying a road perfectly flat because its potholes are shallower than the legal range for how wide the road may be.
Judge smoothness against an actual surface finish and the Earth loses badly. A tournament phenolic ball (Aramith's published figure) has a surface roughness of about Ra 0.03 µm — a thirtieth of a micrometre. Earth's relief, scaled to the same ball, is a gouge 88.6 µm deep — about the width of a human hair, and roughly 3,000 times coarser than the ball's real polish. Run your finger over the scaled Earth and you would feel the Himalayas as a ridge and the Pacific trenches as a canyon. The Earth is not smoother than a billiard ball. It is far rougher.
The slogan: smoother than a billiard ball, but not round enough. The check: round enough — and not smooth. Both adjectives flipped, from one mistake: the ball's lone published tolerance bounds its diameter, not its finish. Apply it to roundness, which it actually governs, and the Earth passes. Borrow it for smoothness, which it does not govern, and you certify a planet "smooth" by a number that has nothing to say about smoothness — while a real ball, polished to a sub-micron finish the rules never bothered to demand, leaves the Earth in the dust.
The underlying numbers in the famous version are all roughly right. The conclusion is still backwards. That is the whole job of this venue: a claim can be built from true figures and still lie, if it measures the wrong thing with the right ruler.
These are recomputed live, in your browser, from the defining constants. The same arithmetic runs offline in research/smoother-than-a-billiard-ball/verify.mjs (17/17 checks). Cited sources are listed below the table.
Sources.
· WGS84 ellipsoid (a = 6,378,137 m; 1/f = 298.257223563): NIMA TR8350.2, World Geodetic System 1984.
· Mount Everest 8,848.86 m: joint China–Nepal announcement, 8 Dec 2020 (Kathmandu Post).
· Challenger Deep 10,935 ± 6 m: Stewart & Jamieson, "Revised depth of the Challenger Deep…," Deep-Sea Research I, 2021.
· Ball diameter 2¼″ ±0.005″: WPA Equipment Specifications.
· Surface roughness Ra ≈ 0.03 µm: Aramith / Saluc published figure for tournament phenolic balls.
· The claim in its original form: Phil Plait, "Ten Things You Don't Know About the Earth," Discover / Bad Astronomy, 2008; careful rebuttals by "Possibly Wrong" (2011) and others.