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Course: High school geometry>Unit 9

Lesson 2: Cavalieri's principle and dissection methods

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 $\text{height}$ and the same $\text{width}$ at every point along that height, they have the same $\text{area}$.
Wait a second, you already know this! Check it out.
What is the area of the rectangle?

What is the area of the parallelogram?

According to Cavalieri's principle, these figures have the same area because they have the same height $\left(6\right)$ and the same width $\left(4\right)$ as each other at every point along that height.

Cavalieri's principle with varied widths

Sometimes a figure has different widths at different heights. Cavalieri's principle still works.
According to Cavalieri's principle, which of the following figures must have the same area?

Try this

What is the area of the following figure?

Want to join the conversation?

• I put 9 pi as my answer for the last one and it said it was wrong
• Put 28.274 which is an approximation to 9 pi and its correct.
• How can we be sure that two figures have the same width at every point along the height?
• 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.
• 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
• 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
• maybe it would help more by putting the formila to the problem
• 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
• 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.
• 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?
• Just type "pi"