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Common Core Math - Geometry: High School Standards6 chapters | 25 lessons

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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.

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.

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.

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!

What's more, if you were to take a samurai sword and chop a plane from each at exactly the same height, the two planes would have exactly the same area. Note that this only works if the heights are the same and the areas of the bases are the same. After all, the area of each plane changes as you move up and down the pyramid or cone. However, according to Cavalieri's Principle, as long as you cut from the same height, you'll have the same area in each slice.

Want the math to back this up? The formula to find the volume of a cone is pi times radius squared times height divided by three. Remember that the pi times radius squared finds us the area of a circle, or the base of a cone. Meanwhile, the formula for the volume of a pyramid is length times width times height all divided by three. Again, length times width is the formula for the area of the base. In other words, pyramids and cones both follow the same formula of base times height divided by three for volume. If the base sizes are the same of each, you could substitute them and get the same result!

But wait, isn't this all sort of stating the obvious? Well, in a way, yes. However, Cavalieri's Principle is an important stepping stone for further parts of math. Here we assumed that each of the layers of our shape had the exact same measurements. Once you go further in your study of math, you'll see that this changes. In fact, the entire field of calculus can be described fairly accurately as figuring out how to measure the volume and surface area of changing layers within a shape. Without the information laid out in Cavalieri's Principle, we wouldn't be able to make sure that the information we gained from calculus is actually correct.

In this lesson we learned how **Cavalieri's Principle** influences every volume formula in terms of area and height. In short, it states that two shapes have the same volume if their heights are equal and their cross-sectional areas are the same. More specifically, cones and pyramids with the same base area and height have the same volume, while cylinders and rectangular prisms with the same height and base area also have the same volume. We also learned that while this may seem pretty obvious to us on the surface of things, it actually makes a huge difference down the road with calculus and other advanced maths.

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Common Core Math - Geometry: High School Standards6 chapters | 25 lessons

- How to Use The Distance Formula 5:27
- Perimeter of Triangles and Rectangles 8:54
- Perimeter of Quadrilaterals and Irregular or Combined Shapes 6:17
- Circles: Area and Circumference 8:21
- Area of Triangles and Rectangles 5:43
- Area of Complex Figures 6:30
- Volume & Cavalieri's Principle 5:26
- Volume of Cylinders, Cones, and Spheres 7:50
- Volume of Prisms and Pyramids 6:15
- Go to Common Core HS Geometry: Measurement

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