Visualizing the Volume of a Sphere Formula | Deriving the Algebraic Formula With Animations
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- Опубліковано 27 вер 2014
- tapintoteenminds.com/3act-mat... In a previous 3 Act Math Task, students watch a short video that shows a cone pouring water twice into a sphere with the same radius/height to fill it to the top. Because it takes two cones to fill one sphere, we can use the volume of a cone formula as a starting point to derive the volume of a sphere. We do this visually using animations in Apple Keynote to make connections between concrete and algebraic representations.
Anyone here from India?
Me 😢😂
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I never knew that two identical cones are equal to the sphere of equal height and equal radius. That is very useful to know. As a 3D modeller, I can imagine squashing a cone to make hemisphere.
I think my eyes just got blessed
+Ratkovski Adajet hahaha! Glad to hear it!
Yes I think
@@britbarbie1699 what is ratk.....😅
This is great. Presented beautifully slowly and carefully so you can actually follow it. Love it.
Beautiful, concise, clear presentation. EXCELLENT!
I wish all mathematical formulas could be explained in such an intuitive way- Many many thank you's
So glad you found it helpful!! Happy Mathing!
Kyle...
Your animations are excellent.
A useful, and orthodox method of deriving the formula for volume directly....
Draw your sphere, centre (0,0).
Allow sphere radius to be r.
Select a value of x to the right of (0,0).
Erect a perpendicular (perp) of height y.
Rotate that perp about the x axis to form a disc.
Allow that disc to have width dx.
The incremental volume of that disc is its area A = pi*y^2 multiplied by its width dx....
dV = pi*y^2*dx
The perp height y is related to x by the classical equation of a circle...
y^2 + x^2 = r^2
make y the subject...
y^2 = r^2 - x^2
It will follow that...
dV = pi*(r^2 - x^2).dx
To determine the full volume of the sphere, integrate that last equation -r to +r...
V = integral of pi*(r^2 - x^2).dx between -r and +r
V = pi*( r^2*x - x^3/3 )
Insert the limits.... -r and +r
V = pi*( r^3 - r^3/3 - (-r^3 + r^3/3) ) = pi*( 2*r^3 -(2/3)*r^3 )
V = pi*r^3*(2 - 2/3) = pi*r^3*(6/3 - 2/3) = (4/3)*pi*r^3
V = (4/3)*pi*r^3
If, later, you want to get the surface area, simply differentiate the volume function, having identified that... dV = A*dr, so A = dV/dr
dV/dr = A = 4*pi*r^2
Best...
Troya.
TroyaE117 Certainly cool! My focus was on trying to make it as understandable as possible (without calculus!) :)
Very interesting approach though!
+Kyle Pearce Thanks Kyle Pearce and Troya for sharing a geometrical and calculus method of deriving the volume formula of a sphere.
This was so simple and well explained. The visuals did great too. Thank you.
Thanks a million. Appreciate the feedback!
The proof using integration is just as beautiful as this
Neeraj Nambiar I’m sure it is! I’m not sure it is as accessible as this one though for many people.
This is the best video in math that I've seen! you make it look so easy and simple, thank you !!
Dear Sir
God bless you for sharing your knowledge.
I'm a retired engineer trying to fill any gaps in my head.
Lol.
I think that you're the answer.
Smile.
I'm going to subscribe and view all of your work.
Much respect.
Great channel.
Fantastic explanation. Thanks a lot.
Mark Hatton glad it helped!
Amazing how simple and yet effective this was
Oh WOW, this is mind blowing in my opinion. As simply as you put it, it's still amazing. I'll definitely use your method in my lesson planning in the future.
Thank you.
+Razan Shammas Thanks so much for your comment. Glad you found it useful!! :)
Yes!! This is what every teacher needs to teach. Awesome job.
WOW!! This was so coooool to learn. Thanks for such a clear explanation and clear visuals! My 6th grader was asking me where does the 4/3 come from, and now we both know! :)
Amazing!! :) Thanks for enjoying the beauty of mathematics!
This channel deserves more subscribers ...Great video.I will pray for you!
This was simple and absolutely great.
Thanks for making me understand the proof of the formula for the volume of a sphere. This was interesting.
Anytime! Glad you're enjoy it!
So simple but well explained! Thanks a lot!
No problem! Thanks for the comment!
Loved
this video. It was clear and easy to follow. After all these years of teaching, I have not seen such a clear explanation. Thank you!
Thank you for the kind words!
I am 63. I really appreciate this. Thank you.
I appreciate the feedback ❤
Incredibly helpful! Thank you.
I always wondered why the volume of the sphere is calculated by that formula, and when I started to see the video where you put water from the cone into the sphere, I immediately understood what you were gonna show me. My head exploded instantly. Now I understand, thanks so much!
So glad to hear that you found the video helpful! Awesome sauce!
Simply amazing. I wish our world has more teachers like you. God bless you
So appreciate the positivity!
Why would anyone dislike this video!
This is awesome. Great job (now I don't have to memorize)
Idris Ayantoye thanks for leaving some feedback! Unsure what part of the video prompted the dislikes, but I'd be curious to know. Glad that you found it useful. My focus is exactly what you've mentioned - to avoid memorization and promote understanding. Take care!
Kyle Pearce maybe the dislikes are for the audio, you need to push the volume a bit up :D but I liked the video, is nice and neat, thanks a lot ;)
Rebius very good point! I will try to keep that in mind for future videos. Thanks for the feedback!
@@KylePearceMathlete you can't see the result in the end because it's covered by an advertisement ... that is the reason I am not showing this to my students .... I didn't give your video a "dislike" though, because this problem can be solved if there is any interest to do so. It is a brilliant way to explain the formula but a poor way of expressing it.
Thank you so much for this!! So many videos just tell you the formula, but knowing why is the key!
Amazing! Two cones volume is equal to a spheres.❤
I actually watched this video to find the area a come, and it was so useful!
This really helped with our online class, thank you so much 👍
First time I understand their volume properly thanks and God bless you
Glad to hear you found this helpful!
This is soo useful man, I have to give a explanation about this tmrw and they didn't teach us about this.
glad to hear it was helpful! Good luck!
Great explanation. But how did we find that the volume of a sphere = the volume of two cones with the same height and radeii as the sphere?
EXACTLY.
Hello! Maybe i'm late but... Because of the Cavalieri's Principle. If you chop a sphere in tiny pieces (with area = πR²) and reorganize them in diferent ways, you got the same volume. Reorganizing all pieces in two cones with the vertex pointing each other, you have what is called "clepsydre" (or hourglass).
By the experiment .
Same doubt
First you try to derive Area of circle by cutting small sectors of say 1 mm arc length. Set the sectors in alternative direction to get a parallelogram of pi x r length and r breadth which gives pi r square. In the same way visualise small cones cut from the crust centre with radius r height also r. Cut the sphere into four pieces. By arranging each cone in alternative direction, Each piece volume will come 1/3 pi r cube. Then comes 4/3 pi r cube.
Mind Blowing, Impressive, Astonishing, Marvelous, Fantastic, Amazing, Beautiful and very easy and clear explanation. Thanks a ton for the video. May God Bless You.
Thanks so much for the kind words!
this makes so much sense now thank you so much
After searching so many video finally got the right place
Thank you so much
God bless you
8 years later, still the best video out there explaining this
So appreciate the feedback!
Confirm for 9 years too!
OMG! I wanted to be able to visualize math forever! I can visualize Anatomy easily, but math... was hard for me to visualize. Thank you so very much! Many many thumbs up!
Thanks for the positivity!
Excellent work for all Nations
Glad to hear it! :)
l usually don't comment on videos but this is the BEST video I could find that explains where the formula for the volume of a sphere comes from! This helped ALOT because I do way better when I understand where the math comes from rather than just memorize a formula. Thank you so much for this top notch explanation!
Monica Mejia I really appreciate your feedback! So happy to hear that the video resonated with you as it was my intention to help make this concept easy to follow!
Best video explaining the volume of a sphere I've seen yet! Thanks!
+Antoine Lalande Much appreciated! Thanks for the feedback :)
It doesnt explain the volume, it says that the volume of a sphere is 2 times the volume of a cone without explaining anything
qbwkp I think I recall doing a demonstration showing the 2:1 relationship when the height and radius is the same.
Kyle Pearce True. The video is also well-made, and i like it. I just want to se the proof on paper.
For sure... I'm definitely going more for a visual than a formal proof. I work with students who may have struggled with math at some point or have gaps in knowledge. Trying to bridge that gap...
It cleared my mind and proof where came the volume formula of a sphere.
Love this video.
2r equals to h of the sphere was a bit difficult to understand but later understood it.
😊
I think I've just been enlightened, great explanation!
Perfect animation and explanation sir...
You are simply awesome
I appreciate you!
Since I haven't studied calculus yet, I couldn't understand the proof of this formula but Sir you made it really easy for students like me.
Thanks alottttt!
this is how it should be taught . thank you
one of the most satisfying videos
The effort done in explaining the concept is massive and so is the video presentation and editing ... nice work, can you tell me which software has been used for editing purposes, looks like a charm...
Apple Keynote + Final Cut Pro X
Wonderful video. I would like to see how we come up with 2(cone) =1(sphere) without the water demonstration. What is the mathematical explanation?
Thanks you bro that's really really help me ,don't stop
oh wow! great job man!
Nice, thanks for the video.
Useful it is... Thanks so much for an easy and assertive explanation..
it makes sense now. thank you!
This is best method to explain maths and geometry, and am sure this method will break tanoo in learners that maths/ geometry is very difficult subject. It's teacher's duty to discover right method to explain the subject, and think this is the most effective way. Thank you very much for the video and wish you best of luck in your endeavours. Namaskar.
best explanation. makes a lot of sense
You have a really calming voice
Glad to hear it :)
Imagine his voice reading a bedtime story. 😛😀
Hello I’m in fifth grade and this was a very good way of understanding a complicated formula
This is so helpful! Thank you!
It's really wonderful!
Wow what a wonderful and amazing explain. Thanks you.
Great explanation. Thanks for making maths clearer...
Happy to help!
very concise, thank you
Excellent explanation....It changed my view towards cone n sphere..
Thanks a lot sir
i just loved it..thanks a lot for the efforts
Syed Iqbal No problem! Hope to get some more up and online soon!
Wow,, thanks alot for the useful information ,, can i ask what program are you using?
Hi I'm Bangladeshi.very nice and easy delivered....so many thanks.
Unmatchable knowledge
I want to thank you for your this good explanation
Nice work Kyle!
Thank you for sharing.
Wowww!!
it was great, thank you..
Which software can I use to make videos like yours?
Excellent explanation and accompanying graphics
Excellent. Thanks a lot.
Oh, I love this!
Excellent work
Wow.. wonderful...many many thanks
Thanks a lot sir. This video been proved a deeper understanding to me. Now i can easily remember the formula of vol of Sphere..Yayyyy !! :)
Vivek Joshi Awesome! AND, if you DON'T remember the formula, just resort to finding the volume of two cones with the same radius and height! :)
Thank you so much
Clear explanation.thanks sir
Really awesome explanation
This is really useful with my maths work. Thanks so much ;)!!!!!
Are you the real Rags
@@oliverkieras6530 are u the real Oliver
I think I've understood now thanx sir.
Genius presentation
Good explanation.
Excellent animation
obrigado pela demonstrações matemáticas
Best way to explain
Well done Kyle; But I want one favor from you can you explain how can we visualize and mathematically come to volume of cone without calculus approach?
Thanks for the comment!
Consider checking out this blog post for more on how we can help students understand where the volume of a pyramid/cone comes from: tapintoteenminds.com/3act-math/prisms-pyramids-3-act-math-task/
It helped me in making my model
Just Amazing
Wonderful explanation
Thankyou so much its very useful to me😀😀😀 I am very happy to learn from your video
Once more thankyou so much
Just brilliant
can you make video how find radius or height if sphere of radius of 3 cm is converted into the cone of radius 6 what will be height of cone ?. height will be 3 .I want to visualize the diagram
Trying to find the proof..Now got it in simple method sir..Tq very much
Osm 👍👍👍 I don't have words to express my gratitude to my new teacher thanks 😊😊 sir
Someone finally decided to make sense out math.
Always trying to find ways to help students see that there is a reason why all formulas work and they can understand those reasons!
if two cone of same radius and same height is able to fill a sphere of same radius then the cone with the same radius is able to fill a hemisphere of same radius then the volume of the cone and the hemisphere will be equal is it possible
Thank you this helped
wow....that was just amazing...i really understood it more clearly and better......
Awesome! Glad to hear it!