The World Pool-Billiard Association Tournament Table and Equipment Specifications (November 2001) state: "All balls must be composed of cast phenolic resin plastic and measure 2 ¼ (+.005) inches [5.715 cm (+ .127 mm)] in diameter and weigh 5 ½ to 6 oz [156 to 170 gms]." (Specification 16.)
This means that balls with a diamenter of 2.25 inches cannot have any imperfections (bumps or dents) greater than 0.005 inches. In other words, the bump or dent to diameter ratio cannot exceed 0.005/2.25 = 0.0022222
The Earth's diameter is approximately 12,756.2 kilometres or 12,756,200 metres.
12,756,200 x 0.0022222 = 28,347.111
So, if a billiard ball were enlarged to the size of Earth, the maximum allowable bump (mountain) or dent (trench) would be 28,347 metres.
Earth's highest mountain, Mount Everest, is only 8,848 metres above sea level. Earth's deepest trench, the Mariana Trench, is only about 11 kilometres below sea level.
So if the Earth were scaled down to the size of a billiard ball, all its mountains and trenches would fall well within the WPA's specifications for smoothness.
However, it should be also be noted that if the Earth is not a perfect sphere. It is an oblate spheroid and should therefore also be tested for conformity to WPA specifications due to its shape. The distance between Earth’s poles is shorter than its diameter at the equator by approximately 43km. The maximum deviation with the respect to the Earth’s average diameter is half that distance, or 21.5km, which is within the scaled up WPA tolerance of 28.3km.
It can therefore be seen that the Earth would conform to WPA specifications for billiard balls if it was scaled down to the appropriate size, and it could said to be smoother than at least some billiard balls that would be permitted by the WPA.
[Thank you to Neil Brennan and Felipe Nievinski for contacting curiouser.co.uk to suggest corrections for this page.]