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### Course: High school geometry > Unit 9

Lesson 2: Cavalieri's principle and dissection methods- Cavalieri's principle in 2D
- Cavalieri's principle in 3D
- Cavalieri's principle in 3D
- Apply Cavalieri's principle
- Volume of pyramids intuition
- Volume of a pyramid or cone
- Volumes of cones intuition
- Using related volumes
- Use related volumes
- Volume of prisms and pyramids

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# Cavalieri's principle in 2D

If two figures have the same height and the same width at every point along that height, they have the same area.

## Cavalieri's principle in 2D

**Key idea:**If two figures have the same

Wait a second, you already know this! Check it out.

According to $(6)$ and the same width $(4)$ as each other at every point along that height.

**Cavalieri's principle**, these figures have the same area because they have the same height## Cavalieri's principle with varied widths

Sometimes a figure has different widths at different heights. Cavalieri's principle still works.

## Try this

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- I put 9 pi as my answer for the last one and it said it was wrong(18 votes)
- Put 28.274 which is an approximation to 9 pi and its correct.(10 votes)

- How can we be sure that two figures have the same width at every point along the height?(19 votes)
- Typically, it will tell you the width if you need it. For example, in a cylinder, the diagram/problem will usually state the radius or diameter. That is the same as the width. This also works for other shapes.(7 votes)

- I claim that 3 * 3 pi is 9 pi. The answer is wrong, they say. So I check the help it claims 9 pi (in letter) is the answer. Bruh(13 votes)
- ik i did this just now and it still wrong i type 9pi and 9π and it says im wrong and when i click the explain button its the same answer as mine still say its wrong lol(3 votes)

- maybe it would help more by putting the formila to the problem(11 votes)
- I dunno, I feel like there is no way that rectangle still had a length of 3pi after it was stretched out. I don't buy it for a second.

Unless that is just the true given length and the example just isnt to size(4 votes)- The diagram doesn't say that the length of the curved line is 3π, it says that the distance from the left edge of the figure to the right is 3π. The curved line is longer of course, but that's not relevant to computing the area.(6 votes)

- On the third question I had to write my answer as a multiple of pi but there was no pop-up so I could put it in there and it's obviously not an option on my keyboard. Is there a way to type pi on my keyboard?(5 votes)
- how does pi affect a wavy rectangle(1 vote)
- it doesn't the length can be an arbitrary constant here pi is being treated as a constant and not significant definations in geometry(9 votes)

- I find the diagram of the last question confusing. I took 3pi to be the distance from the beginning of the wavy figure to the end, rather than the length of the wavy edge.(3 votes)
- That's porbably because 3pi
*is*the distance from the beginning of the wavy figure to the end, rather than the length of the wavy edge. The notation, with a number line with vertical lines, is generally assumed to mean the length of the figure. If 3pi was the length of the edge, such a detail would be stated.(2 votes)

- Could you also find the area of the parallelogram on the second question by doing (6x4/2)x4,since it looks like that the figure is made up of 4 right triangles;(2 upside down and 2 up right)?(2 votes)
- Would Cavallari's principle work if 2 objects had the same height and the same
**average width across the height**(2 votes)