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If the Earth's surface was spherically flat would the sea cover it all?

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vinphetamin | 01:52 Fri 04th Oct 2002 | How it Works
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in answer to the second of your two questions: no.
If you are asking whether if you took the total surface area of the earth and laid it on a flat surface and then took the total volume of seawater and poured it over the top (assuming you had some mechanism to prevent if all flowing off the edges of your flattened earth) would the volume of seawater cover the area of the earth? then I don't know! But In answer to the second part of your question yes I think it does!
I can only assume that you mean if the earth was a "perfect" sphere, would there be enough sea to cover it, then the answer is obviously yes. Its depth would depend on the diameter of the sphere, which would be somewhere between the deepest ocean and the highest mountain.
It would depend on the depth of the 'sea'. There would certainly be enough water to cover the entire of the Earth's surface (especially if you melt the Ploes). But if there was land high enough they'd stay dry, which in all fairness is the same reason why the continents as we know are dry-land even though the Earth isn't flat. I don't know if I made any sense....
To expand on BenDToy's explanation, I did hear a Steve Wright factoid that, if the surface were flat the depth of the ocean would be approx 200m over the entire planet.
Another Wright-ism: "Sponges grow in the ocean. That just kills me. I wonder how much deeper the ocean would be if that didn't happen."

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