Volume & Cavalieri's Principle

Volume & Cavalieri's Principle
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  • 0:03 What Is Cavalieri's Principle?
  • 1:13 Applying the Principle
  • 2:37 Taking It Further
  • 4:15 Why This Is Important
  • 4:50 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

Working with volume in geometry would be a lot harder without Cavalieri's Principle. In this lesson, we see how area and height help us compare the volumes of changing shapes, and we can test our new skills with a brief quiz at the end.

What Is Cavalieri's Principle?

I want you to think of two random shapes. They can be stars, circles, rectangles, triangles, or any other two-dimensional shape. The only catch is that these two shapes have the same area. Now, stretch these shapes out into three-dimensional shapes with precisely the same height. Suddenly, you've got two prisms or cylinders. Believe it or not, these two shapes have exactly the same volume. This idea that two shapes have the same volume if their heights are equal and their cross-sectional areas are the same is called Cavalieri's Principle and has some pretty surprising implications for the study of math.

But wait, how can this be? Think about it like this. When you are calculating the volume of a prism or a cylinder, you are essentially multiplying the area of the base times the height. Sure, for a rectangular prism it may be length times width times height while for a cylinder it may be pi times the radius squared times height, but in any event, you always return to that basic idea of area times height. Cavalieri's Principle states exactly that. In this lesson, we're going to see how this works, as well as how it applies to other shapes like cones and pyramids.

Applying the Principle

Let's take a look at this in action to see how this works. Let's say that you're making a stack of pancakes for everyone, but each stack has to have exactly the same volume. Unfortunately, you have particular guests, and each wants their pancakes to have different shapes. As long as the shapes have the same area, then you just have to make sure that the stacks have the same height to get equal volume. Luckily, because the same sized ladle was used to portion out each pancake, they all have the same area.

According to Cavalieri's Principle, a stack of three circular pancakes will have as much volume as a stack of three triangle pancakes. Why? Remember, the height of each pancake is constant - a pancake can only be so thick, after all. Additionally, because the area of each type of pancake is the same, you can stack them up in equal numbers. Therefore a stack of 3 triangle pancakes is the same volume as a stack of 3 circle pancakes.

But wait, what if someone decides that they want super thin crepes instead? As long as the area of each crepe is the same as the area of each pancake, it doesn't matter! Just make sure you make the same height of crepes as you do pancakes and everyone will have received the same volume.

In fact, it doesn't matter what shape the pancakes take. Feel like a stack of Texas-shaped pancakes? Go for it. Or, maybe you want those that look like a certain cartoon mouse? You can do that, too. As long as the area of each shape is the same and the stacks are the same height, the principle holds.

Taking It Further

Now, imagine if the pancakes weren't in a perfectly straight column. Let's say that they are leaning, sort of like the Leaning Tower of Pisa, but still in a stack. The volumes don't change, do they?

Of course not!

The reason is simple - the base and the height are still the same values. Now, that's probably easy to get with a regular prism, like a cylinder of pancakes. However, the same also applies even across different shapes. Let's say that you were comparing a cone and a pyramid. The area of the base of each is the same, and the height of each is the same. According to Cavalieri's Principle, the volumes of each are the same!

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