### How much water is in the oceans?

Nat Banting posed the following problem on Twitter.

We could just look up the answer on Wikipedia, but to estimate the answer, we need to know the following quantities:

f = Fraction of Earth’s surface covered by the oceans
d = Mean depth of the oceans

Recall that the surface area of a sphere of radius r is

$S = 4\pi r^2$

so the total area covered by the oceans is

$A = 4\pi r^2 f$.

To approximate the total volume of the oceans, we multiply the area by the mean depth.

$V \approx 4\pi r^2 fd$

Now, let’s plug in some numbers. The radius of Earth is approximately 6400 km, and about 71% of Earth’s surface is covered by the oceans. The mean depth of the oceans is 3.7 km. So the total volume of the oceans is

$V \approx 4\pi \cdot (6400 \text{ km})^2 \cdot 0.71 \cdot (3.7 \text{ km})$

which is about 1.35 billion cubic kilometers.

Let’s express this in more familiar units. Recall that a liter is the volume of a cube that is 10 cm on each side. A cubic meter is 1000 liters (since 10 x 10 x 10 = 1000), and a cubic kilometer is 1,000,000,000 cubic meters (since 1000 x 1000 x 1000 = 1,000,000,000). So a cubic kilometer is equal to one trillion liters. This implies that the volume of the oceans is about 1.35 billion trillion liters, or about 350 million trillion US gallons.