# Flat world and water

If the Earth had no mountains or valleys (if the world were flat) would it be covered in water?

So basically, is the mean height of the total surface area of the world (without counting the water, but rather the actual ground) higher than sea-level?

• I assume you mean raising the sea bed to be equal with the height of the other land? If so, obviously the world will be covered with water, because where would it all go? – user16107 Dec 17 '15 at 9:44
• Interestingly, if the Earth were the size of a pool ball, it would meet the regulation smoothness (and even sphericality) criteria. So it's already pretty damn flat. – Whelkaholism Dec 17 '15 at 10:13
• What do you mean by "rather the actual ground"? Do you want to flatten all the land currently not covered by sea or do you want to flatten the whole solid surface? What aspect of the water do you want to conserve whilst doing this - the total volume? – Neil Slater Dec 17 '15 at 10:37
• @Whelkaholism been reading xkcd have we? – James Dec 17 '15 at 14:37
• Link for those not familiar with @James reference: what-if.xkcd.com/46 – cobaltduck Dec 17 '15 at 14:56

If the Earth were the size of a pool ball, it would meet the regulation smoothness (and even sphericality$^{1}$) criteria. So it's already pretty damn flat.

So to do this we need to assume perfection. Wikipedia gives an average radius as $6371\;\text{km}$.

The surface area of a sphere is $4\pi r^{2}$, giving $510,064,472\;\text{km}^{2}$.

Total water on Earth is $1,386,000,000\;\text{km}^{3}$

Apparently, the depth of a volume spread over an area is simply $\frac{Volume}{Area}$.

Giving:

$\frac{1,386,000,000\;\text{km}^{3}}{510,064,472\;\text{km}^{2}} \approx 2.7\;\text{km}$.

So, the Earth would apparently be covered to a depth of around $2.5\;\text{km}$ (some water would still be in the atmosphere I assume).

This seems pretty deep, but a quick Google reveals other people coming up with the same answer so I guess it's right!

$^{1}$ The xkcd linked by James talks about this report, which concludes that the Earth meets smoothness criteria, but not roundness criteria, as the difference in diameter at the poles is greater than allowed for discrepancies.

The link I used divides the difference in diameter by two and so concludes that as the difference each side is less than the allowable error, it DOES meet the criteria.

It's a bit ambiguous; draw your own conclusion. I don't think it affects my answer particularly, given that it's based on completely idealised calculations anyway.

• Thanks for formula edits, I hoped someone would step in :) Must learn how to do that. – Whelkaholism Dec 17 '15 at 11:06
• Just a note: This would require no plate tectonics or significant volcanism. So there would be two kilometers of water between nutrients and sunlight with no mechanism to cause an upwelling from the flat sea floor. No photosynthesis, no oxygen atmosphere. Probably a mix of ammonia and carbon dioxide instead? Maybe methane? This would probably have some effect on the amount of water. – Ville Niemi Dec 17 '15 at 11:23
• what-if.xkcd.com/46 – James Dec 17 '15 at 14:40
• @Whelkaholism if you want to see how the forumlas were done, just hit edit on your post. :) – Draco18s no longer trusts SE Dec 17 '15 at 15:20
• Your formula for volume spread over an area only holds for flat surface (which the earth is not). To find a more correct height of water, you could to solve $\frac{4}{3} \pi (R + h)^3-\frac{4}{3} \pi R^3\ = V$ for $h$, with $h$ the height of water, $V$ its total volume, and $R$ the radius of the earth (i.e. volume of the earth with water minus volume of the earth without water). Doing this I find $2.7162 km$ and $2.7173 km$ with your method. Well... it does not give very different result, I am kind of disappointed ^^' (note that I still assume that the earth is a sphere). – Kolaru Dec 17 '15 at 15:50

One thing that isn't currently considered in the other answers is the potential porosity (is that really a word?) of the ground. A good example of this would be the Rio Hamza, a subterranean river running under the Amazon that is thought to be larger than the Amazon itself. An awful lot of water sits in underground deposits, varying by geological makeup or the surrounding area, average levels of rainfall etc. So:

It depends

If you imagine taking all the mountains, grinding them to sand and pouring it back into the valleys and oceans until there is nothing at the surface but sand then the water level will be much higher than if you take the same mountains, turn them into house sized marbles and pour them into the valleys and oceans until there's nothing at the surface but marbles. The water will drain into the much larger gaps between the house sized marbles, and your average 'ground' height will be greater due to the less efficient packing of the marbles.

If you say 'no: everything should be perfectly flat' then you have to add all current groundwater and subterranean water reserves to the oceans, and as far as I'm aware no-one has ever really measured those accurately (on account of them being under the ground).

It doesn't really matter though. As the other answers have pointed out, the entire surface will still be covered to a depth where it won't really matter how far down the ocean goes!

## It depends upon what you mean by flat.

If you reshaped all of the solid material into a perfect sphere: Due to the earth's spin, the oceans would move and coalesce at the equator. There would be a world-wide ocean straddling the equator, and two continents centered on the poles. Due to its spin, Earth has an equatorial radius approximately 21 km longer than its polar radius.

Such a planet is unstable. Due the spin the equatorial bulge would reform to match the spin and after major earthquakes, etc. you eventually get a planet shaped more or less like Earth.

If you simply bull-dozed the whole plantet completely flat, i.e., to an ellipsoid (geoid to be more precise), in theory any significant amount of water would of necessity cover the whole earth. For very limited water situations, the "oceans" would come and go as a result of wind, tides, etc. But if the available water is enough to overcome such temporary conditions, the whole earth is water covered. As others have already noted, this is the 2.5 km deep ocean that covers the Earth.

Neither solution is optimal for humankind. Or lots of other species either, even if you ignore the severe climactic changes that would necessarily follow.