Learn how to estimate the number of grains of sand on all of Earth’s beaches. Here's Math Dude's trick to quickly estimate tough-to-calculate numbers.

Today we’re going to learn how math makes it easy to estimate things that seem practically impossible to calculate. In particular, since summer is in full swing, we’re going to take math to the beach and think about the age-old question: How many grains of sand are on all of Earth’s beaches? As long time Math Dude fans may recall, we first learned about using math to make estimates when we watched Secret Agent Math daringly calculate how many breaths of air there are in a sealed room. So why are we revisiting this topic? Because learning to combine your brain with math to make estimates is an absolutely invaluable skill—and it’s a skill that’s only developed with practice. Which is exactly what we’re going to do today.

## Step 1: Make a Plan

Sometimes it’s fun to jump right in and do something without really knowing what you’re doing. And while that kind of spontaneity is great for things like claymation and spur-of-the-moment weekend trips, it’s a really bad idea for tackling math problems. As such, the first thing we need to do today is make a plan for figuring out how many grains of sand there are on Earth’s beaches. I know this might sound like a nearly impossible task, but rest assured that overcoming these seemingly long odds—and the sense of dread they instill in the hearts of otherwise brave souls—is precisely what our plan will do.

The best way to tackle a tough problem like this is to break it down into easier parts. In our case, if we can estimate how many grains of sand there are in a typical volume of beach (say the number of grains per cubic meter), and then estimate what the volume of sand is on all the beaches of the world (say in cubic meters), then all we have to do to find the total number of grains of sand is multiply these two numbers together. Easy, right? Well, okay…perhaps this still isn’t exactly easy. But let’s continue marching bravely forward to see if we can tackle each of these sub-problems on their own, and then to see if this divide-and-conquer approach can help make the seemingly impossible possible.

## Step 2: Estimate the Sand Grains Per Volume

The first sub-problem we need to solve is to figure out how many grains of sand there are in a volume of beach. We could measure this with any unit of volume we’d like—grains per cubic mile, grains per cubic kilometer, or whatever—but those would all be difficult estimates to make since they’re all really large volumes. And since the point of breaking the original problem up into sub-pieces is to come up with easier-to-solve problems, a better plan is to start by estimating the number of grains within a small volume—such as the number of grains per cubic centimeter.

How can we do that? Well, we can just gather up a bit of sand and count the grains. But instead of actually gathering a cubic centimeter of sand and counting each individual grain (which, while accurate, would also be tedious and slow), let’s make a quick and dirty estimate by lining up a bunch of grains in a row and measuring the length of the line in centimeters. The number you get will depend upon the type of sand, where it’s from, and a bunch of stuff like that—but if you do this you should get somewhere between 15 and 25 grains per centimeter (please email me and let me know what you get!). So if we assume that an average of 20 grains will fit along each side of a 1 cubic centimeter box of sand, we find that there are about 20 x 20 x 20 = 8,000 grains per cubic centimeter. Which, after a bit of fun converting units, we find is equivalent to 8,000,000,000 grains of sand per cubic meter.

## Step 3: Estimate the Volume of Sand on Earth’s Beaches

Now that we know how many grains of sand there are in a cubic meter of beach, we need to figure out how many cubic meters of beach there are in the world. To find the volume of the Earth’s beaches, we need to know the typical width and depth of a beach and the total length of Earth’s coastlines. In truth, this is a complicated problem since not all beaches are the same width or depth and not all of Earth’s coastline is covered by beaches. But the name of the game here is to come up with a ballpark estimate, so let’s see what we can figure out.

In truth, I’m really not sure what the average width of a beach is, but it seems reasonable to estimate that a typical beach is about half a football field wide—or approximately 50 meters. As for the average depth, it seems reasonable that a typical beach is something like half as deep as it is wide—since deeper than that doesn’t really seem like “beach” anymore—which is about 25 meters.

As for the length of Earth’s coastlines, I’m going to assume that the coastline of each of the classic seven continents is long enough to circle the Earth twice. Why? Because that’s how I’m deciding to make my educated guess. There are lots of other ways you could do it, and my method could certainly be half as big as it should be or maybe twice as big as it should be, but I can live with that uncertainty. The trick is to come up with some method for calculating a good educated guess for the number you’re trying to find (this is where a bit of art comes into estimation), and the particular method I’ve chosen does exactly that. Which means that, since Earth’s circumference is roughly 40,000,000 meters, there are approximately 14 x 40,000,000 = 560,000,000 meters of coastline on Earth. So the volume of Earth’s beaches is about 50 x 25 x 560,000,000 = 700,000,000,000 cubic meters!

## Step 4: Calculate the Number of Sand Grains

We’ve now estimated that there are about 8,000,000,000 = 8x10^9 grains of sand per cubic meter of beach, and that the Earth contains roughly 700,000,000,000 = 7x10^11 cubic meters of beach. Which means that we can find the number of sand grains on Earth’s beaches by multiplying these two numbers together. When we do that, we get 8x10^9 x 7x10^11 = 5.6x10^21. So Earth’s beaches contain roughly 5,000 billion billion—aka, 5 sextillion—grains of sand. Which, needless to say, is a HUGE number!

So that’s the answer we were looking for…and that’s how math can be used to make estimates that help you make sense of the world. If you’re feeling like a lot of this was just guessing, you’re right! But it wasn’t just wild random guessing—it was smart educated guessing based upon what we know about math and the world. And that’s precisely what making estimates is all about!

## Wrap Up

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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Beach image from Shutterstock