# Water Displacement Method & Calculating Density

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• 0:01 The Legend of Archimedes
• 1:30 Volume by Discplacement
• 2:40 Calculating Density
• 4:00 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Measuring the volume and density of irregular objects can be difficult. But there is a way to do this that doesn't require much work at all. In fact, all you need is some water, a scale, and a calculator to help you out.

## The Legend of Archimedes

A very long time ago, there lived a really smart man named Archimedes. Archimedes was so smart in fact, that one day, the King of Syracuse came to him for help. The king had commissioned a jeweler to make him a gold crown, and while the crown looked perfectly fine, the king didn't trust the jeweler. He believed that the jeweler had mixed in a cheaper metal (like silver) to make the crown than what he had been commissioned for, but still charged the king the same expensive price.

Unfortunately, there was no way for the king to prove this, which is why he went to Archimedes. He figured if anyone could help him solve this problem, it was the very smart Archimedes! And while Archimedes wanted to help the king, he couldn't figure out how to determine what the crown was made of without destroying it.

So Archimedes thought about this problem, but he just couldn't come up with a solution. Then, one day he decided to clear his head by taking a nice long bath. He filled the bathtub as full as it would go, and as he stepped into the tub, he noticed that some water spilled out over the side. So he stepped further into the tub and saw that even more water spilled out.

Archimedes suddenly realized that he had found his solution. And legend has it that he was so excited by his discovery that he immediately fled from the tub and ran through the streets naked shouting, 'Eureka!' He took his news straight to the king, and when they tested the crown, they found that the jeweler had in fact cheated the king by using a less expensive metal, which did not end well for the jeweler!

## Volume By Displacement

So now you're probably wondering how Archimedes was able to determine what the crown was made of just by taking a bath. Well, Archimedes knew two important things that helped him come to this realization -- first, that silver is not as dense as gold, and second, that an object's density, mass, and volume are all related to one another. Specifically, that the density of an object is equal to its mass divided by its volume.

When Archimedes stepped into the bathtub, he realized that his body displaced or pushed out a certain amount of water from the tub, which is what caused it to spill over the side. Most importantly, the volume of the water displaced is equal to the volume of the object displacing it, in this case his foot. If he also knew the mass of his foot, then he could easily calculate its density because any time you are only missing one variable in an equation, you can solve for that unknown with the other known variables.

In our story, Archimedes was not interested in the density of his foot, but rather the crown. So the king and Archimedes took that crown, immersed it in water, and measured the volume that it displaced. Once they had its volume, they measured the crown's mass and were able to calculate its density from these two values.

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