How to Solve Archimedes’ Volume Puzzle
Have you ever heard the story of Archimedes and his famous “Eureka!” moment? Want to know how to use his discovery to measure the volume of a pumpkin?
Jason Marshall, PhD
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How to Solve Archimedes’ Volume Puzzle
Half a lunar cycle (that’s two weeks) ago, we spent some time waxing poetic about pumpkins and the delicious pies they bestow upon us. And we spent some time figuring out how we might be able to semi-accurately measure the volumes of pumpkins so that we can figure out how much pie we’re going to be able to make (since we’d never consider using canned stuff, right?).
The method we came up with was based upon the idea that pumpkins are shaped rather similarly to something called an ellipsoid (which basically looks like a sphere that’s been squished here and there … just like a pumpkin). Once we figured out how to calculate the volume of an ellipsoid, we knew how to calculate the volume of a pumpkin, too. But—small problem—pumpkins are often weirdly (mis)shapen, which means that using this method and its rigid formula can be tricky.
As we’ll soon find out, there’s a better way to do it. And as luck would have it (since it makes for such a good story), this way of calculating volume is so exciting that it caused the Greek mathematician Archimedes to run down the street naked shouting, “Eureka!” a few thousand years ago. While it probably won’t have quite the same effect on you, I think you’ll nonetheless agree with me that it’s very clever.
So, what is it? And how does it work? Let’s find out.
Archimedes’ Volume Puzzle
As the story goes, the famous and brilliant Greek mathematician Archimedes was presented with an important problem to solve. A king named Hiero (the second, if you’re keeping track) wanted a new crown, so he gave a bunch of pure gold to a goldsmith and told him to make something pretty and crown-shaped out of it. The goldsmith obliged and got to work.
A while later the king received his new crown. Being a suspicious sort of person, he couldn’t help but wonder if the goldsmith was completely honest or not. The king knew how much the gold he gave to the goldsmith weighed, so he could weigh the crown to make sure it weighed the same. But, he realized, simply ensuring equal before-and-after-weights wasn’t foolproof. After all, a dishonest goldsmith could have easily pocketed some gold for himself and then replaced that weight of gold with an equal weight of some cheaper metal.
The king was flummoxed. He knew there must be a way to tell if he had been stolen from, but he couldn’t figure out what it was. So, he asked Archimedes for help.
How Did Archimedes Solve His Puzzle?
Like many people, Archimedes apparently did his best thinking while sitting in a nice tub of warm water. Because, as the story goes, this is the point at which Archimedes got into the bath. Actually, let me rewind for a second. Before he ever got into that fateful tub, Archimedes realized what he needed to do to resolve the king’s dilemma: He knew that the density (which is a measure of how much something weighs per unit volume) of pure gold is different than the density of gold mixed with some other metal. So Archimedes knew that if he could somehow find the density of the crown, he could compare it to the known density of pure gold to check to see if the king had been bamboozled.
If he could somehow find the density of the crown, he could compare it to the known density of pure gold…
Now, to calculate density you need to know both the mass of something and its volume. Archimedes knew the mass of the crown (it was the same as the mass of the pure gold the king started with). So in order to calculate the crown’s density, he only needed to find its volume. But while it’s easy to find the volume of a cube or a sphere or even a spheroid, it’s really hard to find the volume of something bizarrely shaped like a crown. He could melt it down into a cube, but that would kind of defeat the king’s goal of having a crown made.
So, Archimedes was stuck. And then he stepped into that fateful bathtub. When he did, he noticed that the water in the tub rose. As he sat down further in the water, it rose even more. Archimedes realized that if he dunked the crown in the tub, the water level would also rise … and it would rise precisely in proportion to the volume of the crown (since that’s the amount of water being displaced). If he carefully calibrated his tub of water, he could figure out the additional volume corresponding to each millimeter rise in water level. And thus, he could dunk the crown to find its volume, use this number to calculate its density, and solve the king’s problem.
How to Calculate the Volume of a Pumpkin
Upon realizing what he had discovered, Archimedes leapt from the tub, forgot to grab a robe, and exuberantly ran down the street yelling “Eureka!” telling the world what he had figured out. According to the story, Archimedes confirmed the king’s suspicion and found that the density of the crown was different than the density of pure gold. I’m not sure what ended up happening to the goldsmith, although I’m betting it wasn’t good.
But while his fate might not have been so great, the fate of our quest to calculate the volume of a pumpkin—or any other oddly-shapen object—is looking pretty peachy. Because we now know that no matter how oddly shaped our objects are, we can always measure their volumes. All we have to do is dunk them in a carefully calibrated bathtub, and the answers to our questions will be right there for us to see.
Wrap Up
OK, that’s all the math we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
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!
Crown image courtesy of Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.