Volume of Pyramid: why 1/3 in the formula
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- Опубліковано 30 вер 2024
- This video tries to go over an intuitive approach to understanding why the pyramid formula is 1/3 (base) x (height). In particular where does the 1/3 come from.
The general proof for the 1/3 reasoning is in this video:
• General Proof: Why Vol...
If this video helps one person, then it has served its purpose!
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If anyone is interested in the "general proof" for all pyramids and not just the "cube" case as in this video, then I posted it here (no calculus, all geometry and basic algebra): ua-cam.com/video/F952z0McAhw/v-deo.html
Thanks, this explanation was so eligant that I managed to understand at just 2:58
Happy to hear that the video explanation helped!
I understood the algebra, but this intuitive understanding is far more informative!
Glad the visual intuition helped! Happy learning!
In other words, one of the 6 little pyramids have a volume of half of the full cube, therefore the volume of a full cube would mean you'd have to double the volume of one of those 6 pyramids. That is, 2 x 1 little pyramids which = a third of the volume of the full cube (6 little pyramids together)?
The full cube is made up of 6 pyramids of the same volume. The key was that the height of the cube is 2x taller than the pyramid you are looking for. We know that volume of the cube is BASE x HEIGHT or all the 6 pyramids combined. So we have: (Volume of Cube) = 6 (Volumes of Pyramid). Now solving for "Volume of Pyramid" gives you the desired 2/6 = 1/3 we were looking for. Hope that helps. Happy studying!
@@1mjourney Thanks, does what i wrote make sense to you?
@@domdom9496 First, what you are doing by asking if you understand it correctly, is AMAZING! To repeat what one is trying to understand is the key to learning! I am very happy to get your comment (or comments like it). I hope that I will be able to reply to such comments, but slowly the channel is gathering more speed and it is harder to keep up with comments. To answer you question: in short, no. 1 of the little pyramids is one sixth of the full cube. I encourage you to draw the breakdown of the cube when you are going through the video and see it until you can make the connection intuitively! Cheers and welcome to the 1mjourney.com
triangle is half of a square
a pyramid is one third of a cube
Happy learning! Cheers.
great explanation .Will you please explain the volume of ball?
Great idea! I'm assuming you are referring to the volume of the sphere? Not sure when time will allow me, but I'll put it down on my list of videos to make for the future. Happy learning and welcome to the 1mjourney.com
Hi! Could you proof using same idea (without using calculus) why the volume of a cone is 1/3 of a cylinder? algebraically Thank you!
Hi Flavio Daniel. Thank you for your comment. You are correct that proving the volume of a cone is typically done via calculus. It is much easier to do it that way. You do not really need calculus to do the cone (although you must know that the volume of any pyramid is indeed 1/3 base x height). From your comment, I’m not sure if you are just asking me if it is possible or if you would like me to do a video on that?
@@1mjourney Hey! Thank you very much for your reply! A video will be amazing! But don't know if you have time for that! Hehe. I am brushing all over youtube but there is no good content about it, most of youtubers just apply the formula without explaining why, as you did with the pyramid. Thank you very much for your time and generosity of sharing high quality math content. I am just started following. Thank you!
@@flaviodanielbergozza7552 Hi. I’m happy that the video has helped you. The goal of this channel is to help as many learners as possible. The video I made was for intuitive understanding. Can the same be done with the cone, yes. If you are convinced that the pyramid is 1/3 of the prism, then extending intuitively is possible. Not sure I can whip out a video in the short term, but I will try to think about putting it out there. Welcome to 1mjourney.com …. Happy learning
@@1mjourney thank you very much! I will be following your channel. Cheers!
@@flaviodanielbergozza7552 Hi Flavio. I just posted the video on the cone. I hope this helps. Here is the link: ua-cam.com/video/9YypvVUlUG4/v-deo.html
I’m trying to improve my math so this is so informative! Followed your channel already, can’t wait to watch your videos again! Thanks sir, sending peace and love from Hong Kong!🎉❤
Amazing to know that the video has helped you! Happy learning! Great to have someone in Hong Kong watching :) ... welcome to 1mjourney.com ...
In addition to h = H/2, I think you also need W=D=H. You initially mentioned that it is a cube with equal lengths, but your drawings are showing different dimensions. The tips of the 6 pyramids meet at a single point at the center of the cube.
Always fun to see a comment here! The sides are indeed equal of the original cube. Note that the reason why you are seeing the further drawings as of different size as I double up the pyramid to show the result. If you are really keen in understanding the general case of this proof (unfortunately the video for this is much longer) then do not hesitate to take a look: ua-cam.com/video/F952z0McAhw/v-deo.html
A student asked me this same question today in geometry class. Now I have an answer. I wonder if anyone makes a math manipulable with 6 congruent pyramids that fit together in a cube. Seeing is believing, right? Thanks so much for creating this amazing explanation.
Hi! Happy to hear that the video has helped. This video takes a special case for the pyramid that allows a visual understanding of where that 1/3 comes from. If you ever want to see the full proof (with visuals) then you can take a peak at my general case video for the pyramid. Super happy to hear that the video is helping students, teachers and learners alike!
Why can you assume that the side pyramids are identical in general to the two original pyramids (from the bottom and from the top)? That is true only when W=H.
Hi! You are correct. This video captures the special case to simplify the understanding of a geometric proof. However, if you would like to see the general case you can. I pinned the general proof video. It is longer however :) … Happy learning and thanks for the comment!
I loved the way you teached! Even my 13 year old sibling understood. I wonder if you can also teach us why the frustum of pyramid also has "1/3" in its Volume. More power to you sir!
Hi Reynaldo - you asked twice for this :) ... ok ... I made the video for you. Here is the link: ua-cam.com/video/kzATapIES3w/v-deo.html hit a like, subscribe and pass on to others! Cheers, 1mjourney.com
Thank you, try to find this explanation with visualization in brazilian portuguese, but didn't find anything about it!
Glad it helped! Welcome to 1mjourney.com :)
This explanation really helped me to understand thank you very much
Great to hear the explanation helped! Happy learning!
damn ur explaination was so smooth. thank u so much. God bless u
Glad it helped! Happy learning ...
Yes, but I have never seen a non=-clculus method for establishing this fractional (1/3) relationship. In fact I know no calculus method that establishes the general fact either. This was why I looked at your video. Thanks for your response. Paul Ryan
Hi Paul Ryan. Thank you again for sharing your thoughts. As I mentioned, I enjoy critical feedback. Your comment has planted a seed in my mind. Using Calculus is not ideal for intuition. It may do the trick on proving something, but since this formula is introduced in elementary school (for most) then Calculus will not do the trick for younger kids. I will try to put this on my to do list. Many thanks for sharing your thoughts!
Dear Paul Ryan. I have just posted the general proof of this 1/3 relationship. The video is much longer than this intuitive explanation posted here. I did not use calculus. If you would like to go through it, here it is: ua-cam.com/video/F952z0McAhw/v-deo.html ... I hope this will ease your mind about the 1/3 relationship for pyramids. Best of wishes, 1mjourney.com
Thanks! The video was super helpful! But just one question, why do we need to split the height in two, and why can’t we just use the equation Base x Height instead on Base times 2 heights in the end?
*Instead of not instead on*
Great question! The Base x Height would consider the entire cube, but what I tried to do is to divide the big cube into 6 symmetrical pyramids. In order to do that, I had to divide Height into a half with the tip of the pyramid touching that mid point and having a height of “h” and not the height of the big cube “H”. I hope that helps. If not, then let me know. Happy studying! 1mjourney.com
@@1mjourney thanks!
thank youu, visual proof help me alot when dealing with formula
Very happy to hear that the visual has helped! Happy learning and welcome to the 1mjourney.com
👍
Glad you found it useful!
dostum çok güzel anlattın !!!!
Glad it helped! Happy learning!
Thanks loved your explanation
Hi! Great … if the explanation helped then the video has done its job :) … Cheers!
half height hh. Middle of cube.
h=half height in cube. 😊
Cheers! Happy learning!
Very nice indeed. What software are you using for the drawings?
Hi Peter. I draw by hand, but I do use Notability as my note making app. It used to be free, but I don’t believe it is free anymore. Cheers!
That was very clear! How can do we get the interior angles of different polygons or its Formula? Thank you.☺
Hi! Glad the video has helped. The interior angles will depend on the polygon so there is no one set formula for angles, but given enough information about a polygon one can always break things down to trigonometry :) ... Happy learning!
@@1mjourney Thanks.
I still dont understand😅😅
Stay positive and do not give up on trying to comprehend the reason! Cheers!
I don't get how you got 1/6?
Take a look at the video one more time and notice that there are 6 pyramids in the entire cube. Since we want only 1 pyramid so we must take the factor of 1/6! Hope that helps!
what was the use of splitting the sides into halves and marking it? 2:06
Hi! Good question. It was just to point out that the midpoint inside of the cube is projected from those halves. Looking at the video again now, I should have pointed that out. My apologies if it was not clear.
@@1mjourney no its ok, thanks for answering, i understand it now!
Thanks you so much sir❤❤❤
You are very welcome! I’m glad the video has helped … welcome to 1mjourney.com
BEST VIDEO
Great to know it helped! Cheers!
hey what software are you using to write this?
Hi. Notability is the app that I am using for drawing/writing/etc. Unfortunately it is no longer for free. Notability has grown and is now subscription based. I still got it when it was for a few bucks and have all the features grandfathered. Hope that helps! Happy learning.
@@1mjourney thanks
It iiss amazzzing!
Great to hear it helped!
What program did you use to draw? I love how clear and precise your mathematical drawings are, especially the ability to instantly change drawn lines to straight lines.
HI Sarah! I use "Notability". It is an amazing app for notes. It has many built in features (like the one you mentioned). I wish they would sponsor the channel :) ... unfortunately at present no sponsors, but I'm very happy that you found the video useful. Cheers and Happy Learning!
What is the answer
Hi Maretis. What question are you trying to have answered? If it is why 1/3 is present in the formula, then the video shows this by an intuitive restrictive example. If you are trying to find the answer to the general proof, then please take a look at my “pinned” video which is much longer. Feel free to let me know. Cheers!
Clear! Ty🙏🏽
Glad it helped! Happy learning!
Thanks. I liked it, well explained.
Happy to know that the video has helped! Welcome to 1mjourney.com
@@1mjourney Actually someone asked a teacher and the response was not satisfying. So I was bothered and really wanted to know why. Thank you so much. It was perfectly explained. I liked the algebra in it. Live long sir.
@@lanomusambazi-tx2eg Hi Lano. Sounds great! Feel free to share with your teacher and other students. Your feedback is appreciated!
but the side pyramids are not the same as the top/bottom pyramids.
Hi! Yes they are. The constructed shape is a 3d cube. This is a special case that I tried to give an intuition of the proof. The general proof is much longer, but I did post the video for it. Please see the “‘pinned” comment I gave on this video. with the link to the general proof. Happy studying!
@@1mjourney Thanks, just found your other video. It is indeed much longer. Will take time to finish it.
@@zhongtom2625 Sometimes it takes a little bit of time to learn something new :) … I hope you find it useful. It is interesting how many people actually do not know why we have a 1/3 in the formula. Cheers!
Thank you, that really helped👍
Best 👍 teacher
Happy to hear that the video has helped you! Welcome to 1mjourney.com
But why does the height need to be divided?
Hi! Happy studying :) ... the cube Height (H) = 2h (where h is just half of the cube height. The reason why I did that is so that I can construct the whole cube from 6 pyramids (bottom pyramid, top pyramid, and 4 pyramids of the side). These pyramids construct the whole cube and since we are interested in the formula of the pyramid then we can take the entire cube volume and divide by 6 to get our 1/3 base x height (where this height is half of the height of the cube). I hope that helps. Good luck!
May I ask.. Uhmm is the height of the pyramid half the height of the cube?
@@starrynights5988 Yes, it is for this set up.
@@1mjourney Oooh THANK YOU 😂
AMAZING explanation, thank you so much 😊
Glad it helped! Happy learning :)
Thanks 😊
Happy to hear that it helped! Happy Learning. Welcome to 1mjourney.com
Great job explaining. My 9 y/o son understood this.
HI. It is so amazing to hear that the video has helped your son. The goal with each video is to help just one person, if that happens, the it was worth the time to create it! Happy learning! 1mjourney.com
Thank you so much. Was struggling to figure this out.
You are very welcome. Please note, all my knowledge is build on the amazing educators and learners that have come before me. I just try my best to mirror their passion for teaching! Happy learning and welcome to the channel!
Thanks for the explanation 👍
Glad the video helped! Happy studying! Cheers, 1mjourney.com
Thank you sir, clear explanation and nice drawing
Glad to know that the video has helped you! Happy learning :)
Great video…you made it easy to understand. Thank You
You are welcome! I’m glad the video has helped … feel free to see the general proof for any pyramid (see pinned comment I posted). Welcome to 1mjourney.com …. thank you for supporting.
Amazing. This really helped me dive deeper into the topic simply taught at school. Thanks a lot.
Glad to hear that the explanation helped!
Thank you sir . superb Idea 👍
Enjoy! Hopefully the explanation made sense. Cheers! 1mjourney.com
Understood, very nice!
Great! Happy the video helped. Happy studying!
Thank You so much
Happy to hear that the video helped!
Thank You!
You are very welcome! Happy to hear that the video has helped in some small way. Welcome to the 1mjourney.com