Quick – how much does the ocean weigh?

Let’s say you’ve recently been spending a lot of time at home with a small child. You foolishly told them you can’t play right now because you’re busy doing important work calculating the weight of stuff. This has sparked their interest. And they won’t let go of that question.

Sigh. I don’t know how much the ocean weighs. I couldn’t even guess how many zeroes are in the number.

Or could I?

70% of the Earth’s surface is covered in ocean. I’ve watched enough nature shows to remember that’s about right. What would that surface area be? Let’s assume the Earth is a sphere (it’s not, but it’s close enough). What’s the formula for the surface area of a sphere? I don’t remember and my Weight Engineer’s Handbook is in the office. No matter – let’s assume Earth is a cube instead. I know how to find the surface area of a cube.

I know what you’re thinking – working from home with a hypothetical kid asking dumb questions all day has driven me bonkers. We can’t assume the Earth is a cube. But for this estimate we can. All I’m looking for is a number within a few orders of magnitude of reality. The difference between a sphere’s surface area and a cube’s (with the cube’s edge the same length as the sphere’s diameter) is negligible.

The surface area of a cube is six times the area of one side of height H, or 6xH². Since our cube’s height is the same as our sphere’s diameter, we need to know Earth’s diameter. Because “Apollo 13” is an awesome movie I know that a spacecraft in low Earth orbit moves about 17,500 miles per hour and takes 90 minutes to complete an orbit. That gives a circumference of 26,250 miles and a diameter of 8360 miles. Let’s call it 8,000 miles because I read somewhere that low Earth orbit is about 50 miles up, the Earth is rotating under the spacecraft and I’m kidding myself if I’m going to assume any precision in this calculation.

That gives us an ocean surface area on our Earth-cube of 70% x 6 x 8000² = 269 million square miles.

I don’t know the average depth of the ocean, but I know most of it’s deep. I’m guessing between one and ten miles because I heard “miles” and ocean depth mentioned together somewhere. Let’s say two miles deep. That gives us a total volume of 538 million cubic miles.

Every weight engineer knows the density of water – it’s 1 gram per milliliter. That’s fresh water – salty ocean water is denser, close enough. I remember it that way because I’m an expatriate Canadian and the metric system is a tenacious thing. No matter, I can convert. There are 454 grams in a pound (thanks, Canadian food packaging!), a milliliter is the same as a cubic centimeter and there are 2.54 centimeters in an inch. Put that all together and you get a water density of 0.036 lb/in³, or 9,181,017,236,653 pounds per cubic mile. Time for scientific notation: 9.2x10E12 pounds per cubic mile.

At a volume of 538 million cubic miles, our ocean weighs 4.9x10E21 pounds.

How did I do? According to the internet, “*Earth’s ocean is made up of more than 20 seas and four oceans, weighing an estimated 1,450,000,000,000,000,000 short tons*” – or 2.9x10E21 pounds.

There you go, kid. My estimate came within 2x the real value. That’s pretty good for starting out with no earthly idea!

This is called “Fermi approximation”, named after World War Two-era physicist Enrico Fermi, who was known for making good approximate calculations with little to no data. It’s a great way to get a quick rough guess before moving on to more precise methods.

My first boss when I was a brand new mass properties engineer used this method to devastating effect. He would send me off to spend days researching, interviewing designers and analysts, drawing sketches and running calculations to estimate the weight impact of a potential design change. When I returned, he would knock off an estimate in a few seconds that would invariably be within spitting distance of my hours of work. It was humbling, and I would ask him why he made me go to the effort. He’d say his method is quick, but now I have the data to back it up. He didn’t say it also made me a better mass properties engineer, by giving me a powerful tool to quickly understand the size of a thing, whether it’s a potential weight impact or the number of hours a task might take – or the answer to a childish question.

If you like hearing about the techniques, methods and experiences of other weight engineers, I encourage you to attend the 2020 SAWE Tech Fair, starting June 22. A wide range of technical presentations, industry seminars and training classes will be featured, all live and online, presented by your mass properties colleagues.