How to Derive The Volume? Hard Geometry Problem
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- čas přidán 20. 04. 2013
- mathematicsonline.etsy.com
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complete explanation for volume of a pyramid here:
pythagoreanmath.com/deriving-t...
Sum of integers squared video link:
• Sum of integers square...
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.
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
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
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.
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!
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
my friend, this visual explanation is the best i have ever seen. Thank you. Keep up the excellent work
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.....
These are excellent and extremely well done. From one math teacher to another, you are a superb educator.
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.
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.
Great explanation with equally good visuals! Loved it and subscribed!
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.
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!
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 :)
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..
Beautifully done. Visuals were super helpful thank you
I love this! Excellent explanation and walkthrough of the proof. :)
First time I understand how their multiple of 1/3 came thank you out soooooooooooomuch
Great video for anyone. Thank you very much for making this!!!
That was a very concise video, thanks!
What an amazing video with marvelous explanations! Thanks a lot!
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?
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?
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.
Wow this is amazing. Like seriously. Brilliantly simple. The visualizations aid soo much. This could've saved me hours of pondering in school.
thanks a lot! this really helped me understand, I kinda really didn't get the point... amazing explanations
*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?
Beautifully done.
I assume this works with any shape base that focused with straight line to one point at the top.
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.
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?
how long did it take to create this video? I am interested in creating similar videos.
Your link to the sum of squares video isn't there.
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⚡💥💣💥
thank you very much it explains everything i was stuck at!!!
There is no link in the description! :(
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!!!
may I ask on 6:18 part why did they exchange the denominator? thank u
To get away from indeterminación inf over inf.
This is a nice video !
What software did you use to make these animation?
finally, a detailed explanation.
Cool explanation man!
From stepsis to step pyramid what a journey I cleared
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 :)
Very good amazing explanation
Your videos are awesome and very informative and are on a different level from most explanations, Thank You.
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
Great Explanation!
Awesome video!
NICE,IT HELP ON MY STATE EXAM,THANKS! :)
This is an excellent video.
Excelente dedução . Parabéns
Thank you, you saved my time
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
I don't understand why you transfer 6 with n^2
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! 👏😮
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
Is there a special class that teaches this? If your using a book, what book shows this?
Excellent Stuff!
wonderful!! Keep going bro
looking over this again, length = prism # (L/n) only applies when each number prism is twice as long as the next one down. Is this honestly true for all pyramids?
+Andrew f Here's the general infinite sum that works for an arbitrary base/height. lim(d->∞)"sum as "i" goes from 0 to ∞" (bh/d)(1-i/d)^2. From this we can derive the general formula, bh/3.
You're my shepherd. I just killed the subscribe button.
Keep posting.
So in any equation... If i know that n closes to 0 if i increase it to infinity... I can just ignore it?
+klaik30 Yes you can. That is called a limit.
5:25 dont you need a Summation on the left?
amazing explanations
amazing stuff thank you
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
Legendary video
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.
heigth of the prism?
Heigth?
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
Please do one for Triangular Pyramid
This is amazing
Why switch denominators???????
Es difícil entender perfectamente lo que dice ya que no hablo muy bien inglés pero aún así entiendo gracias a sus gráficos
Está excelente
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.
I appreciate you taking your time to explain this on youtube. Good video!
mind blown. Thanks.
Thank-you sir!
Brilliant! Thanks for this. :-)
Magnificent! I enjoy your pace and thorough explanations!
1:58
There are 2 spellings of height on screen, and it is spelt height, and not heigth.
Mind blown! But how did the greeks do this? Amazing!
An outstanding proof. Who came up with it?
amazing , thank u so much !
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.
this can be applied for any cone
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 lesson. this is how you spell height :)
I always thought that the volume of a pyramid was derived from cutting a prism into three pyramids
same
Isn't it b^2h\3?
Great videos
thank you vvvveeeerrrrryyy vvvveeerrryyy mmmuuuccchhh....genius...
thanks for the video
Why do you call the individual rectangles "prisms"? I thought a prism was a triangular solid.
Nah, rectangular prism is the correct term for a shape like this
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
It's nice, but is confusing because you have used the same variable many a time like for no. of prisms, and for the slices, etc.