Cavalieri's Principle in 3D | Volume of a sphere |

To improve your problem solving skills, go to: brilliant.org/...
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Finding an equation for the volume of a sphere using Cavalieri's Principle ( assuming we already know the equation for the volume of a cone)
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/ think_twice
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Any further questions or ideas:
Email - [e-mail]
Twitter - / thinktwice2580
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Programs used:
- Cinema 4D
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Music:
" [[FREE]] LoFi Type Beat - "4 am" "
• [[FREE]] LoFi Type Bea...

[@skiggywiggy8386]

Ah! I finally understand how they came up with that. Understanding why the formulas work makes math more fun

[@dareleven]

wow it's unimaginable how animation can be way more effective in conveying complex ideas such as this compared to teaching with merely words, numbers, and drawings alone. I really hope one day I could learn animation too!

[@JustMe-ui9bv]

It's fricking beautiful! I love this content.

[@conoroneill8067]

Hmmm... Is there an explanation for where the formula for a cone comes from? Everything else is great!

[@ThinkTwiceLtu]

In this video I assumed that we already know the formula for the volume of a cone, however it can also be found using Cavalieri's principle. if anyone is interested i can make a follow up video on how to find the volume of a cone.

[@kingkeroro2525]

yes please!

[@NoNameAtAll2]

@@ThinkTwiceLtu yes, would be great

[@beenaalavudheen4343]

Yes please!

[@denisfuryaev8166]

We want

[@46pi26]

Does this suffice as a rigorous proof of the statement that math=art?

[@ineedken5400]

I am so with you on that statement! Math needs to be taught creatively, not this BS that has been going on for centuries.

[@Joffrerap]

rigurously, it's more like math ∈ art.

@@Joffrerap Nah, it just shows that math and art have a non-empty intersection.

[@michaelren8110]

spectacular video

[@avocadosauce6840]

PLEASE do a video on the Taylor series.

[@farisakmal2722]

I second that

[@devanshgarg31]

Yep

[@natasadjurdjevic3533]

3blue1brown has a great video about it

[@Saltofreak123]

i think i found a way to show the intuition behind the taylor series, now only need to get good at animating. working on it right now :P

[@adrianhdz138]

Avocado Sauce Why did I think of Taylor Swift when I read this?

[@djb903]

Very smooth! Subbed instantly

[@DarkLightning96]

Found this channel a couple of days ago and it's already one of my favorites. Really shows how beautiful maths can be :)

[@faith3174]

i've seen so many proofs for the volume of a sphere but this here has to be one of the most elegant proofs ever. great job!

[@mr.mathbook1412]

It is good to see that mathematics in the high level,easy to understand for students and also understand math very deeply .Thank you for such this chanel ,keep going on.

[@dominicjung4950]

I've also known this method for many years. But out of many many ways of proving the volume of sphere this is without a doubt the most elegant and beautiful way in my opinion! :D

[@seanwrfps]

wow, the animation is so clean and crispy. nice job!

[@nicuradu911]

Made my day! Content quality

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It's great to see this Chanel getting sponsorship!

[@farisakmal2722]

THINK TWICE, I MISSED YOU... BRILLIANT WHY DO YOU HAVE TO MOLEST SUCH A CUTE CHANNEL

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Channels _want_ sponsors.

[@mather468]

@Gabe Catalfo What the fuck is that comparison? What does it have to do with the sponsorship relation?

[@dnickaroo3574]

These animations which derive formulae visually are very helpful.

[@PaulFisher]

This also indirectly shows another cool circle thing, which is that if you have two circles with the same center (a flat donut, like your cone and cylinder), and you take the distance from the inner circle to the outer circle along a line tangent to the inner circle, and make a new circle with that radius, the area of that circle will be the same as the area of the flat donut. Or, to put it in the terms of the Car Talk puzzler where I originally learnt this, if you measure a carousel from edge to edge with a line that just meets the inner hole, it will take the same amount of paint to cover the deck of the carousel as it would a circle with that diameter.

[@theskycuber4213]

There should be a double like button on KZbin, extremely elegant explanation.

[@deepakvyavahare7436]

Please keep posting I’m a math enthusiast and I adore these videos.

[@MagicGonads]

It only makes sense to me if the principal is that = cross sections on the same bounded height => equal volumes but not the converse. IE equal volumes does not necessarily imply that the cross sections are always equal when bounded by the same height. Which is what you said. But when I first watched this video I for some reason read it the other way, and had a bit of a debate to myself, oops.

[@pedronunes3063]

Yes, example a paralelopipedid (I don't know if I wrote it right) with volume 1

[@antoniolewis1016]

Glad it makes sense to you! You are correct that shapes with equal volumes may have different areas at cross section, but if they have equal areas at all cross sections, then their volume will be equal.

[@quahntasy]

Such a spectacular video, please do one on Taylor series.

[@Goku17yen]

Here from Flammy!! Love the ambient atmosphere you create lol :D

[@6lbs._onion]

Great visualized explanation as always! :^D I wish they teaches these back when I was still in school... Blindly memorizing the formula? More like "you'd better shove these pile of symbols down your throat, without us explaining it thoroughly why it is like that then vomit it out on the test or else you won't get past jr.high" memorize... x] Where I live... Let's just say the education system is *very* underdeveloped and extremely outdated...

[@rastafarayy]

Cavalieri is older then the way we do integrals now

[@illumilenny8543]

Simple, beautiful, to the point. This is quality content.

[@maxng7916]

This is getting better than mindyourdecisions.

[@gentko2159]

underrated channel for such mesmerizing content

[@iVideoCommenter]

ThinkTwice and 3B1B. Leaders in Visual Mathematics! :)

[@rajeevaggarwal2937]

Your videos change my underrstanding about formulas and derivation

[@sethbracken]

There’s a lot to like about all your videos, but I really like the color palette you’ve stuck to.

[@ThinkTwiceLtu]

Seth Bracken thank you:) Sometimes I spend an hour or more just picking the colours for the animation

[@Sam_on_YouTube]

Personally, I like the calculus proof of that equation better because I feel it leads to a better understanding of calculis and also of why the formula looks the way it does. But this also nice.

[@jameroth7661]

I'm baffled; the simplicity is magnificent. Well done.

[@manuelhoffmann4470]

I like this. Especially the music! 😁

[@DerAlfredman]

*Mind Blowing*

[@rahul7270]

Yay! I'm so happy to see a notification from you! Let me watch the video now.

[@06kokoko]

Holy crap…this is mind blowing

[@banderfargoyl]

Nicely done! I can't imagine a better way to find that formula.

[@rodrigo-vl7bi]

Your videos are one of my favorite things on the internet, they are really awesome, thanks for making them

[@ThinkTwiceLtu]

rodrigo thanks a lot:)

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Best math channel !!!

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This is beyond satisfying

[@everr.r]

absolutely beautiful. simple and efficient.

[@cyrill2357]

Well, it’s obvious that your work is frickin beautiful and very much inspiring! If you don’t mind me asking, what animation software is it that you use? I plan on using some visual representation for some school projects (:

[@ThinkTwiceLtu]

Cyrill Thank you! I’m using processing and cinema4d for my animations. And then I just edit everything in a video editing software like adobe premiere pro

[@NeilDeshpande313]

Utterly satisfying

[@liaailia4454]

it's brilliant! thanks for making this vedio. it was really helpful to prepare shcool presentation!

[@MJMECH]

Superb...easy way to learn

[@JorgetePanete]

i love this channel, it really help us

[@nintendogos61]

Cute lil shapes💛

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Elegantiška, paprasta, gražu. Super Think twice

[@ThinkTwiceLtu]

Elena Jonikiene ačiū :)

[@CinnamonyPasteline]

I HAVE A DUMB QUIZ ON THIS TOMORROW AND I DONT KNOW ANYTHING BUT THANKS

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I get nothing but I do like to watch this videos

[@Pedritox0953]

You won a subscriber

[@oussamanhairech5178]

Thank for this , I am really amazed

[@deidara_8598]

"Blindly memorizing formulas", perfectly describes modern math education.

[@ThePharphis]

I don't really see it, tbh. I tutor on the side and remember high school and university quite well, and very few times were we meant to just memorize stuff. There was always some aspect that aided in memory and the methodology was just as important as the answer

[@cosimobaldi03]

OHH SHEEIT NEVER REALIZED THAT BUT IT'S SO PERFECT

[@goncalosanhudodeportocarre9591]

You could have kept the cone pointing up the cross section animation would be easier to understand. Still great work love the channel

[@tas378]

This is art I love it so aesthetic

[@freeshavaacadooo1095]

This is such simple integration, you don't even need to name this a principle, this is just common sense calculus.

[@SimchaWaldman]

The more time passes, the more videos about this topic are uploaded.

[@manjulata9680]

Your videos are best

[@eccentricOrange]

Absolutely love it! Seen all your videos

[@wenhanzhou5826]

Wow, this was surprisingly good!

[@yamansanghavi]

Mind = Blown

[@lithostheory]

Nice!

[@emirkandemir58]

beautiful explanation, thanks!

[@Mayur7Garg]

People more interested in this principle should watch the video on Napkin Ring problem by Vsauce. He uses the same principle to explain the concept of a napkin ring.

[@joseffnic3560]

Super cool.

[@HoaPham-yk5wn]

very interesting. thank u

[@InTheBeginningTheUniverseWas]

oh look another maths/science channel I'm subscribed too is sponsored by brilliant.org. . . . didn't see that one coming

[@Snowflake_tv]

Great! I appreciate it.

[@basil9633]

W video , helped with hw

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I love your content and i appreciate your efforts ❤️❤️❤️❤️❤️

[@luizfelipegarcia4676]

Very nice

[@hindigente]

Simply beautiful!

[@beenaalavudheen4343]

Great video! Hope u are feeling better now.

[@ThinkTwiceLtu]

thank you

[@damianmatma708]

01:14 I thought there is a typo in the word "hemisphere" and that there should be "semisphere". But "hemisphere" is also a correct word (it comes from the Greek word "hemi" which means "half" - the same as Latin word "semi"). english.stackexchange.com/questions/416547/whats-the-difference-between-a-hemisphere-and-a-semisphere

[@zergdeveloper9227]

Thanks 👍

[@gazalisameer4173]

I am your new subscriber. And i think your channel is so amazing i cannot describe. Please tell us which software do you use for creating these slinky animation. I am a mathematics teacher from india and i too use animation from adobe after effects but they are not quite amazing plus they are time consuming Please tells us the name of application you use

[@lualalsa]

Superb.

[@AdityaKumar-ij5ok]

My textbook also only taught the formula for volume of sphere, without telling how it comes in first place!! Even I as kid of 9th standard have clear intuition of the volume of solids in between two parallel planes, I was thinking about getting it under my name😃😃, but it already existed with name of another mathematician, thanks

[@kaustubha7371]

The one showed up again 💙 What do you think about the Atiyah Riemann Hypothesis?

[@SVP-uy9qb]

This channel is great!

[@lampofthestreet]

Best channel

[@ethangilchrist3534]

These videos are amazing, you've earned that 200:1 like:dislike ratio

[@anthonycannet1305]

How do we know the radius of the one is equal to its height? I’m assuming it has to do with it being made based off a hemisphere but can you show more specifically

[@minionologyforyou301]

These videos are so great!

[@ThinkTwiceLtu]

Minionology for You! thank you:)

[@randomcosmonaut8150]

I always wonder why the surface area of a sphere is the derivative of its volume. can you explain why? great visual interpretation by the way edit: is there any other way to get the surface area formula beside derivative?

[@musicalBurr]

Lovely!

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another amazing video!!👏👏👏👏 keep going

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Make a video on Bassal problem

[@jabo5008]

Beautiful!

[@funkysagancat3295]

awesome video :) keep up with the good work!

[@abrarshaikh2254]

Holly shit!!! I am stunned!

[@GenericInternetter]

if you do this with a 4d hypersphere, the 3d cross-section grows linearly

[@markdenversorino4136]

Nice paradox

[@vanshgarg9560]

You are the best

[@SxB1337]

Amazing!

[@cavver3523]

Pretty good!!!

[@natasadjurdjevic3533]

Do the cone volume

[@moonwatcher2001]

Gorgeous

[@shubhrajit2117]

Oh, Awesome!