# If the Earth were a smooth spheroid, how deep would the ocean be?

At the moment there are deep seas and high mountains. But imagine that the land elevation of the Earth is equal everywhere. How deep would the ocean be in that case?

• – Spencer Aug 15 '17 at 3:29
• The newer question Will the oceans swallow all of the land? is likely worth related viewing, asking a about the feasibility rather than the values. – JeopardyTempest Oct 16 '18 at 17:05
• Also the consideration in my answer about some of the dirt becoming additional mud may be a realistic consideration that may need to be made to the answer here as well. – JeopardyTempest Oct 16 '18 at 17:05

An approximation can be obtained quite simply by dividing the volume of water in the oceans by the surface area of an ellipsoid with a smooth surface representing the idealized Earth in your question.

The volume of Earth's oceans, seas and bays is $1.332 \times 10^9 \text{ km}^3$.

The equatorial radius of Earth (semi-major axis of the spheroid) is $a = 6378.1 \text{ km}$. The polar radius of Earth (semi-minor axis) is $c = 6356.8 \text{ km}$.

The surface area of the oblate ($c < a$) spheroid is:

$$S = 2 \pi a^2 \left( 1 + \frac{1 - e^2}{e}\tanh^{-1} e \right)$$

where $e^2 = 1 - \frac{c^2}{a^2}$.

Which gives us $\approx 0.51 \times 10^9 \text{ km}^2$.

Dividing the volume of the oceans by this results gives us $\approx 2.6 \text{ km}$.

Note: Earth is not a sphere. An ellipsoid is a better representation of our Earth. Nevertheless, the answer to your question would have been approximately the same had I used a sphere instead, as suggested in the title of your question.

• Do you think that because of all the mountains and seadephts the sizes you used (6378 and 6356) are wright? – Marijn Feb 3 '16 at 20:27
• You can try and add the water of all lakes and lagoons, as well as the water in rivers and ice in glaciers and mountaintops, and see how much a variation you get. I am not much familiarized with the Snowball Earth hyphotesis. – carnendil Feb 3 '16 at 20:54
• @userLTK -- It's the other way around. If this hypothesis is correct, ice sheets up to 3 km thick ice covered the continents, but not the oceans. The oceans would have been covered by a much thinner layer of ice, possibly with partially open water near the equator. – David Hammen Feb 7 '16 at 14:15
• @DavidHammen Thanks, and that makes sense. Ice would build up on land but perhaps stay liquid in the salty oceans, with, as you said, a thinner ice layer on top. Sea level must have dropped perhaps as much as a mile in that case. – userLTK Feb 8 '16 at 4:40
• Even with a smooth Earth, the oceans wouldn't be a uniform depth. For the same reason Earth is an oblate spheroid, the Earth + water would be a slightly larger oblate spheroid. The ocean would be shallower at the poles and deeper at the equator by the same ratio as the Earth's axes, so 1/290. – kwknowles Apr 11 '16 at 3:43

510,100,000 square kilometers of surface area, and a total of 1,386,000,000 cubic kilometers of water gives you a 2.717 kilometer column of water across the whole planet if it was billiard ball smooth, but the same basic shape.

• I should note that there will be some variance, in the range of metres due to tidal and rotational effects, and very very minor variances due to local thermal conditions. – Ash Aug 16 '17 at 11:16
• Hopefully, you mean ideally snooth rather than the much rougher surface of a billiard ball. Earth is already smoother than a billiard ball. – Spencer Feb 11 '18 at 1:16