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
4-minute read
Episode #263

Gold CrownHalf 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.


About the Author

Jason Marshall, PhD

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.