Nearest order of magnitude

August 31, 2017
1.1.2 State the ranges of

PKD-Award-CoversThe latest installment in Randall Munroe's XKCD "What If?" series is called Paint the Earth and it is amazing. One of Munroe's readers wanted to know "Has humanity produced enough paint to cover the entire land area of the Earth?" and Munroe uses this as a springboard for explaining Fermi estimation, a powerful, counter-intuitive tool that has applications in many fields.

animation (2) But first, let's think about different ways we might come up with a guess for what the answer will be. In this kind of thinking—often called Fermi estimation—all that matters is getting in the right ballpark; that is, the answer should have about the right number of digits. In Fermi estimation, you can round[1] all your answers to the nearest order of magnitude:

Let's suppose that, on average, everyone in the world is responsible for the existence of two rooms, and they're both painted. My living room has about 50 square meters of paintable area, and two of those would be 100 square meters. 7.15 billion people times 100 square meters per person is a little under a trillion square meters—an area smaller than Egypt.

Let's make a wild guess that, on average, one person out of every thousand spends their working life painting things. If I assume it would take me three hours to paint the room I'm in, [2] and 100 billion people have ever lived, and each of them spent 30 years painting things for 8 hours a day, we come up with 150 trillion square meters ... just about exactly the land area of the Earth.

How much paint does it take to paint a house? I'm not enough of an adult to have any idea, so let's take another Fermi guess.

Based on my impressions from walking down the aisles, home improvement stores stock about as many light bulbs as cans of paint. A normal house might have about 20 light bulbs, so let's assume a house needs about 20 gallons of paint.[3] Sure, that sounds about right.

The average US home costs about $200, 000. Assuming each gallon of paint covers about 300 square feet, that's a square meter of paint per $300 of real estate. I vaguely remember that the world's real estate has a combined value of something like $100 trillion, [4] which suggests there's about 300 billion square meters of paint on the world's real estate. That's about one New Mexico.

Of course, both of the building-related guesses could be overestimates (lots of buildings are not painted) or underestimates (lots of things that are not buildings[5] are painted) But from these wild Fermi estimates, my guess would be that there probably isn't enough paint to cover all the land.

Orders of magnitude exercise example 1 | Pre-Algebra
Orders of magnitude exercise example 1 | Pre-Algebra ...
Order of Magnitude
Order of Magnitude
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