High school geometry
- 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
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
and the same at every point along that height, they have the same .
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
and the same width 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?
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(18 votes)
- How can we be sure that two figures have the same width at every point along the height?(17 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.(6 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(11 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)
- What is the radius of the moon(4 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)
- Just type the two letters p and i which make the term pi.
Hope this helps!
Have a BLESSED day!(2 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 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(3 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.(4 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)