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Using Displacement to Calculate the Volume of an Object

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  • 0:02 Archimedes' Discovery
  • 1:18 Volume by Displacement
  • 3:22 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.

Determining the volume of an object is easy if that object is uniform and has a regular shape. For objects that are irregularly shaped, using the edges is not so straightforward, so instead we can measure the volume by displacement.

Archimedes' Discovery

About 2,000 years ago, a smart guy named Archimedes was charged with an important task. The king of Syracuse had had a crown made for him by a jeweler, but he didn't trust the jeweler. The king suspected that the jeweler had used a cheaper metal for production (like silver) but had still charged the king for the expensive gold he had been commissioned to use.

Alas, the king had no way of proving this. But he knew how smart Archimedes was, so he asked him to prove it for him. Archimedes wanted to help the king, but he couldn't figure out how to determine what the crown was made of without destroying it.

Poor Archimedes thought... and thought... and thought. But he couldn't come up with the solution, until one day when he decided to clear his head by taking a nice long bath. As he stepped into the full bathtub, he noticed that some water splashed out. So he stepped further into the tub and saw that even more water came out.

He knew right away that this was the answer to the king's problem. He was so excited by his discovery that he immediately fled from the tub and ran through the streets naked shouting, 'Eureka!' He ran straight to the king with his news, and when they tested the crown, they found that the jeweler had in fact cheated the king by using a less expensive metal.

Volume by Displacement

You may be wondering how Archimedes was able to determine the crown's composition by simply stepping into the bathtub. Well, like I said, he was a pretty smart guy. He knew that silver is not as dense as gold, and he also knew that the mass and volume of an object are related to its density.

Specifically, the density is equal to the object's mass divided by its volume. Archimedes made the connection in the bathtub when he realized that as he put himself in, he displaced a certain volume of water from the tub, causing it to spill over the side. If he measured the amount of water that overflowed, he could determine the volume of the object displacing it, and if he measured the mass of the object, he could, therefore, determine its density.

Archimedes and the king were able to determine the density of the crown by measuring its mass and then immersing the crown into a tub filled with water, thereby measuring its volume by displacement.

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