Cavalieri's principle in 3D | Solid geometry | High school geometry | Khan Academy

Cavalieri's principle tells us that if 2 figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.
View more lessons or practice this subject at www.khanacademy.org/math/geom... .
Khan Academy is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We offer quizzes, questions, instructional videos, and articles on a range of academic subjects, including math, biology, chemistry, physics, history, economics, finance, grammar, preschool learning, and more. We provide teachers with tools and data so they can help their students develop the skills, habits, and mindsets for success in school and beyond. Khan Academy has been translated into dozens of languages, and 15 million people around the globe learn on Khan Academy every month. As a 501(c)(3) nonprofit organization, we would love your help!
Donate or volunteer today! Donate here: www.khanacademy.org/donate?ut...
Volunteer here: www.khanacademy.org/contribut...

[@jonathannagel7427]

Interesting way to not have to use Reinmann sums & sigma notation! I’m a physical/visual learning person who probably could have told you from a young age that if you sliced a hot dog several ways, it would still displace the same amount of water... as I’m trying to teach my niece some advanced ideas in elementary ways, this is very helpful!

[@activepg3d713]

Its a real skill how he can talk and use his mouse to draw

[@livethefuture2492]

its not a mouse, its a one of those sketch pads.

[@betelgoose9712]

Thank you so much. The way I was being taught this made it sound so complicated. This makes perfect sense.

[@tejamitsrivastava7047]

god i am no way near the class this is for but iam still learning it in summer break

[@ZennExile]

I am not coming up with a scenario where this might be useful. It feels a bit like every slice of water in bowls of identical volume will be equally wet... if that makes sense.

[@joycewong4689]

4:46 the sphere looks like an onion lol

[@thattimestampguy]

Infinite Chocolate 🍫

[@arianaazmir4321]

What about cylinder that is pressed at one end? Does it still have the same volume as the original cylinder?

[@arianaazmir4321]

Will the cross sectional area be the same? Because the the circular will gradually chang to illeptical as it reach the pressed end. So can the cavalieri priciple applies to toothpaste tube shape? (Cylinder with one end pressed)

[@GoatOfTheWoods]

more polygons, though

[@chunchen3450]

Are u Zach Star😁

[@himanshusharma1220]

Did u saw Sushant Singh Rajput metioned you in a post before his death 😓😓😓😓😓

[@muhammadhussainsarhandi9928]

Where is that post?

[@himanshusharma1220]

On instagram search- Sushant singht rajput (read the cation on one of his post where he was doing coding)

[@beachboardfan9544]

This seems obvious, surprised theirs a principle for it

[@seradfb345]

I think it seems obvious because it is displayed in such an insightful way. I came here from learning that adding a multiple of one row of a matrix does not change the determinant. It seemed quite a surprising fact to me and it made clear sense when I saw it visually. Anyway, the most beautiful maths is sometimes the simplest.

[@natehaydin6423]

Item 1- wide top / narrow middle / wide bottom Item 2 - narrow throughout. Both have the same height. How is the volume of both items the same?

[@Shibzzeg]

I think you're forgetting the second rule he mentioned - the cross sections at corresponding heights should have the same area. In your example - the cross sections at corresponding heights are different (and therefore, their areas are different). Imagine 2 tall stacks of CD's with the same number of disks in each of them. You could say that a single CD in the stack is a cross section of this stack. No matter how you skew any of these stacks - their volumes will be the same, because they are composed of identical cross-sections at every height and their total height is the same. Now replace one of the CD's in one of the stacks with a mini-CD. Its thickness is the same as the regular CD's, so the total stack height will remain the same, but its surface area is smaller. In this case, the volumes of two stacks will not be the same, because you replaced one of the cross sections in one of the stacks with a smaller one, so you made the stack narrower at one point (this is the same as Item 1 in your example)

[@richbuilds_com]

This doesn't hold true if you have say a pyramid and another inverted pyramid of the same volume. The tip of the first obviously has a different cross section to the base of the other. ...Unless there's addendum rules to Cavalieri's principle that state the two shapes must be in the same orientation?

[@richbuilds_com]

I'm a software developer by trade. I'm always looking for the edge cases ;-)

[@Raynover]

What is the software you used for the 3d objects?

[@rashedulislamseum7936]

But how could cut this kind of figure??

[@Rancrom]

I can’t work the sphere out, doesn’t each cross section have to be smaller than the next? Otherwise would make a cylinder wouldn’t it? Perhaps I misunderstand that part of the principle.

[@Rancrom]

Actually, I think I understand. It’s because it’s cutting at an infinitesimal point rather than a physical cut so the shape at that point should be mirrored exactly. Is that right?

[@algotn]

@@Rancrom Yep that's right.

[@downwindone]

Section-n body1 == Section-n body2

[@DoesMahBlockLookBig]

is this somewhat similar to a Jacobian?