How to Derive The Volume? Hard Geometry Problem
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- เผยแพร่เมื่อ 8 ก.ย. 2024
- mathematicsonl...
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complete explanation for volume of a pyramid here:
pythagoreanmath...
Sum of integers squared video link:
• Sum of integers square...
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The geometry formula videos come from my curiosity to find out where they come from, I search for clues around the web and I share it with you on youtube.
Keep going
Where should I click to view that too?
I watched this because of my curiosity , so please keep going. Best wishes.
I don't know how you do it, but.. all of your Videos are AMAZING!!... the VISUALS are so important in representing the Intuition.. and YOU HAVE MASTERED that.. Thank YOU!!
Unique Style of Teaching is that a stock image of a teacher? I’m sold
There's a way to cut a unit cube into 6 congruent pyramids, each with a base of 1x1 and a height of 1/2. Each has a volume of 1/6 because it takes 6 of them to make a cube. You can stretch said pyramid to make other square pyramids. (Give it a height of "h" and you have to multiply the height by 2h since it's currently 1/2. A base of l by w means you multiply the volume by l and by w. Thus, you get a volume that is (1/6) x 2h x l x w = (1/3)hlw, without use of limits or large sums or any heavy algebra. This seems a more intuitive approach to me, if you're talking about rectangular pyramids. (Not so with other shapes of bases.. but for an initial introduction. . . ) Thoughts on that?
Naomi Anderegg very simple intuitive explanation!
Naomi Anderegg Very smart and simple soluition.But this solution only applies to CUBE
"There's a way to cut a unit CUBE into 6 congruen...."
It takes additional trick for brick.
Good explanation tough, Naomi.
+Stephanus Kusuma The additional trick you mention, is simply SCALING. When you scale an object for example factor 2 in one direction, the volume of that object also increases with factor 2. Since a brick is simply a cube scaled differently in different directions, the same explanation holds: 3 pyramids fit into 1 brick.
+Gijs van de Lagemaat then do it ur self if u think that and if he's doing a bad job than u do it ur self
He made another vid. just for this.....
This is a great way to introduce calculus. This video, surface of a sphere video, volume of a sphere video. Gives an intuitive sense of what Riemann sums and limit states are actually doing before you start memorizing the integral tricks.
The best video -- not skipping a single step! Very visual!
This is literally calculating an integral from by definition
I was expecting some sort of geometry proof but the video was just an integral and I’m like bruh
It’s also teaching calculus to an audience who might not understand calculus.
any pyramid with any simple closed region R as its base and a has a height of h would have a volume of |R|h/3 by means of integration
@@oni8337 It's true, even if you used a cone, or a projected image of Casper the friendly ghost, scaling down to a point.
Rarely seen such a perfect and clear explanation of a mathematical formulae derivation. The voice-over combined with the amazing creativity of the videos explaining visually the words of the voice over - this is totally out of the world. I have subscribed and I will be seeing all your videos and revising my math. Thank you for your videos.
wow, I've just discovered this channel. This is amazing! thank you for doing all these amazing videos. providing the proof of a concept is essential. and also isn't an easy task. Really happy about finding it. :D
This is scary. I was just working out how to define this same formula using integration and I look over at my phone and this video is at the top of my recommended feed 🤯😱
Well, that'd happen to one person out of so far 251k math students watching this video. Nothing scary, just probability :)
Keep up the good work man. I'm re-learning math and got curious about why this works, and I got to understand it from your video.
Glad to hear this, thanks!
Very clever demonstration
my friend, this visual explanation is the best i have ever seen. Thank you. Keep up the excellent work
What an amazing video with marvelous explanations! Thanks a lot!
Please keep making explanation videos like these :(( my textbooks and math teachers seem to ignore the fact that we students need to know WHY, HOW and WHERE these genius formulae come from, too.
These are excellent and extremely well done. From one math teacher to another, you are a superb educator.
Geometrically if you push the volume up against the wall on the centreline , copy it, flip the copy, line it up on the other side, it makes a rectangle prism and the holes on the sides are 1/2 the volume.
Proof of this is left to the reader as an exercise.
Great explanation with equally good visuals! Loved it and subscribed!
Thank you very much for the proof! My math teacher said that whoever will lecture the proof the volume of the pyramid to the class will earn 5 points more on the upcoming exam.
Just use calculus. One simple integration of a constant and you get the formula
Calculus!!!!???????
Sir! Do you want to see flames shoot out my ears! We don't need no stinkin' calculus⚡💥💣💥
I watched the video as many times as finally I understood. Thanks for the great job. love this channell will share it with friends too.
What is interesting is that you don't even need to calculate the sum of squares in 4:47. That's because we know that sum of polynomial sequence a_n x^n + a_{n - 1} x^{n - 1} ... a_0 of degree n will always result in polynomial of the degree n + 1. That means that the limit is already convergent, as quotient of two polynomials of the same degree always converge when approaching infinity and not only that, but the limit only depends on the coefficient of the highest degree. We also know that an polynomial of degree n can be exactly defined only by n + 1 points (all with different x coordinates). Knowing all that, we can interpolate the sum of squares if we have 4 points (to get polynomial of degree 3), but as we only need coefficient of the highest degree, there is no need to interpolate all polynomial. We can calculate de coefficient of the highest degree very simple, here is an "recursive" algorithm:
1. Take n + 1 points of polynomial of degree n with different x coordinates.
2. Take differences of the consecutive terms, forming a new sequence out of them.
3. Repeat point "2." until there's is only one number.
4. The coefficient of the highest degree is that number from point "3." divided by n! and divided by the power of n of the interval between consecutive x coordinates ((x_{k + 1} - x_k)^n), assuming that it is constant.
In our case we have:
1. Points are (1, 1^2), (2, 1^2 + 2^2), (3, 1^2 + 2^2 + 3^2), (4, 1^2 + 2^2 + 3^2 + 4^2)
2. The differences would be:
2^2 + 1^2 - 1^2 = 2^2 = 4 | 3^2 + 2^2 + 1^2 - (2^2 + 1^2) = 3^2 = 9 | (4^2 + 3^2 + 2^2 + 1^2) - (3^2 + 2^2 + 1^2) = 4^2 = 16
9 - 4 = 5 | 16 - 9 = 7
7 - 5 = 2
4. So the coefficient is 2 / (3! 1^3) = 1 / 3 and the first term is 1/3 n^3
If you do that algorithm with more than n + 1 points, you will see that at certain point you will get sequence of constants. It works because difference between consecutive terms will always eliminate the term of the highest degree, for example, (n + 1)^2 - n^2 = 2n + 1, 2(n + 1) + 1 - (2n + 1) = 2. If you track down how the final value is calculated without simplifications, you will basically get the definition of the n-th derivative. Note that the n-th derivative of polynomial of degree n gives the derivative exactly, no matter what interval you choose. It only works with polynomials, whereas with other functions you get only approximate value.
finally, a detailed explanation.
*I HAVE A DOUBT!* Can anyone prove why the volume of a pyramid does not depend on the position of the top vertex when it is placed in a plane parallel to the base and only on the height of the top vertex and the base area?
Great video...I really enjoyed thanks :)... Though I got to point one mistake in it... On minute 7:10... It is said that "as it approaches infinity the number becomes so small that it actually becomes equal to 0..."... that's actually not possible as infinity is not a integer but a concept and Maths says that it is not possible to divide by 0 out infinity... They both are concepts used in limits where we get the number to be so small that we actually take it as a 0 but it will never be a true 0, it will be 0.000....001. That's the way we use to know what happens when we deal with infinity... Anyway it has been a great video that fascinated me and I only wanted to point that out... Thanks for the vid :)
Thank you, very helpfull to understand the concept of integral, any chance to generate a similar video related to a sphere? using the same approach
really superb......... please continue to make some more. i cant find many more from u in Utube..
Wow this is amazing. Like seriously. Brilliantly simple. The visualizations aid soo much. This could've saved me hours of pondering in school.
This was so helpful but oh my goodness this is so much work
...but good to know the understanding is not beyond your reach. I was taught this formula in the mid 1970's and felt cheated there was no proof given. 50 years later and I can die happy knowing I can actually follow the proof. This is GOLD!!!
These are amazing videos. Your a genius. Keep it up. I know that you have low video views, but deriving information is very rare, I think. These videos are a necessity in the world.
First time I understand how their multiple of 1/3 came thank you out soooooooooooomuch
You're my shepherd. I just killed the subscribe button.
Keep posting.
Your videos are awesome and very informative and are on a different level from most explanations, Thank You.
thank you vvvveeeerrrrryyy vvvveeerrryyy mmmuuuccchhh....genius...
Deduzi a série dos quadrados como:
(1/3)*n^3 + (1/2)*n^2 + (1/6)*n
e o número de blocos numa pirâmide multiplicando-a por 4:
(2/3)*(2n^3 + 3n^2 + n)
mas isso foi num processo totalmente geométrico, ou braçal, retirando-a de dentro de um cubo, tal como numa lapidação, e a “anti-pirâmide”, ou entulho, ou Antimatéria, é:
(2/3)*(4n^3 - 3n2 - n)
Onde somando pirâmide e “anti-pirâmide” teremos nosso Cubo quadruplicado.
Great video for anyone. Thank you very much for making this!!!
Woooow. At first I stopped and digested this like nahh whats he talking bout. Had a flash back then I finished the video and 💥💥💥boom it all made sense! 👏😮
That was a very concise video, thanks!
They give us formulas and make us solve hundreds of equations and problems as a torture, and they completely miss all the interesting stuff. Wish I had better teachers back then
MrOfstring ikr they dont give a damn
Are you really could understand this at 6th grade...? Really?
This is calculus 1 mate, what he did was just an overcomplicated integral.
@@boktampu if they're intelligent enough to use formulas in appropriate contexts they should be smart enough to grasp the basic logic behind how we came up with these formulas
Legendary video
Beautifully done.
Very nice video with a clear explanation.
Can I suggest that rather than writing the (prism #) as ‘n’ you give it another letter instead? This confused me a little as in 04:45 I thought... couldn’t you just cancel the n’s in (n*L/n)^2 ?
It took me a little while to figure out the ‘n’ you gave for the prism # is different to the ‘n’ in the number of slices.
I love this! Excellent explanation and walkthrough of the proof. :)
Thank you, you saved my time
Mind blown! But how did the greeks do this? Amazing!
Beautifully done. Visuals were super helpful thank you
1:58
There are 2 spellings of height on screen, and it is spelt height, and not heigth.
Very good amazing explanation
Food for thought: The formula Bh/3 tells you that the area is exactly 1/3 of a box with the same height and base as the pyramid. Can you cut the pyramid into n pieces, hopefully identical pieces, so that 3*n of those pieces can be used to fill the corresponding box. If so, then this would be a simple geometric proof for A = vol(box)/3 = Bh/3, especially with the software used in the video.
+monty1618 Doing this with a blob-ased pyramid would be very hard.
qbwkp I've thought about it since posting. An easier geometric proof would be to cut 6 pyramids from the cube. The base of each pyramid is a face of the cube, and the apex of each pyramid is the center of the cube.
monty1618 But you cant do this with pyramids that have a pentagonal base
Can you do it for a triangular base?
monty1618 I think so, i think you can do it with pentagons as well, its when the hexagpns come in where i think it gets tricky.
This is a nice video !
What software did you use to make these animation?
When I ask for any proof, My teacher says you have to explore it yourself it is not in the syllabus and just tells the formula directly without any knowldege of the source of the formula
Es difícil entender perfectamente lo que dice ya que no hablo muy bien inglés pero aún así entiendo gracias a sus gráficos
Está excelente
From stepsis to step pyramid what a journey I cleared
which software you have used to make this visually fantastic video...?
you have given a wonderful explaination.
please tell me I am also wants to try this to taught structure design to Architectural Students.
thanks! I used adobe animate
This is amazing
wonderful!! Keep going bro
I assume this works with any shape base that focused with straight line to one point at the top.
mind blown. Thanks.
very good......
thank you very much it explains everything i was stuck at!!!
Awesome video!
great lesson. this is how you spell height :)
Volume of any pyramid and cone, not just square pyramid.
Why did you divide the number of slices of the length the same was as the height?
well n represents numbers. so the the n became representative of all numbers. remember the goal is to infinitely deivide them. so you can't place a number there. so you go ahead and set up a method where you can lim n>infinity. so you can remove all n's
thanks a lot! this really helped me understand, I kinda really didn't get the point... amazing explanations
NICE,IT HELP ON MY STATE EXAM,THANKS! :)
can someone please explain to me how he switched between the n^2 and 6 at 6:15 ? or if this is an algebraic rule then please let me now its name. and thank you upfront👍
this is multiplying any fractions: a/b x c/d = ac/bd = a/d x c/b
@mathematicsonline oh it's that simple! Thanks, man, and great video, BTW.
Thanks for you kind words my friend.
Excelente dedução . Parabéns
This is the beauty of mathematics..
Great Explanation!
So, this is a wonderful way of deriving this, I was curious about this before watching the video and thought that taking an integral of the area of a square l^2 dl giving (l^3)/3, why is this approach incorrect?
This is an excellent video.
It’s easier to understand with the use of Integration.
This is basically integration, though
Bin there. Tried that. Melted my brain into mush. Not going there again.
Thank-you sir!
7:22
so with more steps n approaches infinity and 3/n approaches 0, but it can never be 0 or it is undefined. so why is it possible to sub the limit of 0 into the equation
it's isn't "undefined". and about it "approaching" zero not becoming zero isn't actually true, it BECOMES zero. he said it approaches zero because the more you increase the closer it is to zero, infinity means you increase so much that is becomes 0, not any close number to 0, just Zero.
and even if it was actually a very close number, it would be something like L^2h/3+(2+0.0000000000000000000000001+0.0000000000000000000000001) which would equal L^2h/3+2.0000000000000000000000001 and assuming L^2h/3 equaled, say, 27, you add 2.0000000000000000000000001 to that and it'll be 29.0000000000000000000000001, at the end it will still be just 29. this is a simlifying example but the number that "approuches" zero would be much much more smaller than that you can not even write it, that's why it's "infinite" just like how 9.99999999...=10.
@@MegaMoh That is ABSOLUTELY not true, and it is a very damaging thing to say to someone. It NEVER becomes zero, it approaches it! That's the reason you can't put 0 in the denominator or infinity. Calculus is always approximation, that can become arbitrarily accurate. It never becomes zero or infinity very very important detail. It's the same reason that dx^2=2xdx and not just =2x.
limit as n approaches infinity of 1/n = 0
@@MegaMoh Yes but there's a huge freaking difference. And I'm sorry if this is news to you.
This notation is true lim of 1/n as n approaches infinity=0
YES that is true however, that does not mean that it actually BECOMES 0 if it actually became 0 you wouldn't be able to integrate or differentiate to begin with. It's all just notation what the 'lim n going to infinity =0' means is just NOTATION. In reality when u differentiate dx^2=2xdx you have an infinite amount of subterms you choose to throw away because they do not meet your requirement for precision!
You MUST realize the difference or it will impede your ongoing math studies I would be quite sure of it.
Again it DOES NOT become zero. The limit does.... difference.
notation notation notation.
When you write 0.999...=1 (which is mathematically true) the '...' is notation for limit.
Good video, everything very clear, however I have a question. What if the base of the pyramid was not rectangular? How could you get the formula for your volume?
Cool explanation man!
amazing explanations
I'm confuse as to how and why he divided the length of the base by the high of the pyramid to get the length of each slice base. Can someone explain?
Great video though the word is: height. Spelling and pronouncing it as heigth tends to undermine the correctness of what you're trying to show.
... or by doing the integration of the area applied to the whole height, you obtain the same formula faster and more reliably
Francois Swine and people would ask what's integrals
Just check on google C'mon it's not that complicated understanding fundamental things ... all these videos can be summarised in anything but one principle (which is analysis fundamental theorem by the way, then why not making any effort searchinf about it in order to have this intuition until the end of your life?)
Understanding calculus and understanding why integration can be used to derive an area is actually complicated to someone who hasn't learned it, shockingly... If you asked your 15 year old self to learn all that he would have a hard time. This video gives a much more intuitive proof than calculus could.
The purpose of this video is to promote a geometric intuition of the volume of a pyramid. That's more didactic by not using calculus -- as you said, it'd simplify the whole thing to just one formula, which some people wouldn't even understand.
Ok but he can say at the end of the video that what he did was equivalent to integrating, it would have been useful for people willing to learn that afterwards
An outstanding proof. Who came up with it?
Excellent Stuff!
amazing stuff thank you
how long did it take to create this video? I am interested in creating similar videos.
Love the animations!
As for the maths themselves, regardless of the correctness of the result, is not this “proof” essentially flawed? I mean, isn’t it wrong to equalise a straight line to infinitely small steps or zigzags or semicircle-shaped bumps or anything of the sort?
th-cam.com/video/jUeIjZI32Jg/w-d-xo.html
Knowing the details of this accounts for the smile of eternal repose on the face of so many pharaohs as they lie in state and their ka-spirit arises to the nail of the north......
Magnificent! I enjoy your pace and thorough explanations!
Please do one for Triangular Pyramid
I appreciate you taking your time to explain this on youtube. Good video!
horizontal slices, not vertical slices. But brilliant, thanks
Now this is what I've been wondering for years.
I respect your non calculus Archimedesian approach to Math. (where possible)
5:25 dont you need a Summation on the left?
I always thought that the volume of a pyramid was derived from cutting a prism into three pyramids
same
6:10 How can you get away with switching the denominators?
+FailedNuance algebra, multiplying can be done in whatever order 2 x 3 = 3 x 2 or in this case, (24/5)(27/6) = (24/6)(27/5)
I couldnt understand. (24/5)(27/6)=4,5 and (24/6)(27/5)=5,4 , right?
(2/6)(3/5)=6/30=1/5
(3/6)(2/5)=6/30=1/5
e.g.
12/2 x 6/3 = 6 x 2 = 12
12/3 x 6/2 = 4 x 3 = 12
@@wesleyfurtado8065 uuhh, it's 21.6 in both cases
Is there a special class that teaches this? If your using a book, what book shows this?
I am late , but why did u divide the triangle 😢, i thiught each strip is lomger than the other by 2L/n not by L/n
Your link to the sum of squares video isn't there.
difficulty in understanding
but thanks