@@flower_girl4983 learn the basics first for example the properties of triangle and other basic shapes, then go for average (this video) and finally the difficult ones. That's how you can master the art of learning mathematics
***** Just intuitively, humans have been doing that in math long before calculus. One thing that is cool is that mathematicians were close to inventing calculus back in the ancient historical times. I forgot the details but I think I read it in a Lancelot Hogben book.
Kemerover He didn't use integrals did he? I only watched the surface area of a sphere part. Oh when you said cone I started to think about volumes sorry. Anyway not all formulas require calculus to derive them. Some formulas were derived about 1500 years before calculus. Some of the concepts that are used in calculus were developed about 2,000 years ago.
Ivan Ereiz Yes he just used one of the concepts used in calculus but he didn't use the 'language' of calculus so this is a nice piece of work that everyone who has high school algebra can understand.
Thank you very much! This video is very helpful for students like me who have yet to learn calculus, but still want to understand what they're doing. I can usually come up with my own "proofs" for most formulas, but when it came to spheres I was completely lost. Now it makes sense to take a polygon with infinite sides, just as you do with the circular(ish?) part of cone. Thanks! Really hits home the beauty and creativity of math, especially for a subject that most people assume is dry with no room for creativity.
Beautiful! Raw simplicity & beauty of mathematics presented with clear & concise explanation and graphics. It doesn't get much better than this! Thank you, thank you, thank you!
By far, the most elegant and unique derivation of the formula, without calculus, which makes it understandable to a larger number of students. A mathematical elegance presented in clear and concise graphics and a truly immaculate approach. It can't get any better. Thank you, on behalf of all the students who are not yet introduced to calculus! Beautiful. Subbed instantly!
Damn who came up with this -_- i understand but i would never think of something like this. Imagine you were in a time where all you had was a sphere in your hand and someone was able to think of this AMAZING
manipulating pre existing resources and bending it to fit to a new problem was my introduction to calculus,its not easy,but does not take a genius to do this, fascinating regardless
Beautiful! Just a random question emerged in my mind when I was solving physics problems turn out to has one of the most fascinating explanation about math that I've ever watch. The video is clean and smooth, didn't expect this quality from a 2013 TH-cam Video. Thank you so much!
@@mathematicsonline Thanks for sharing but I just don't see how or why anyone would cone up with this at all? Especially since it's so convoluted and unintuitive. My idea for a proof is 4 pi r squared is 4 times the area of a circle so you can think of a sphere as having four "faces" like a box has four faces. So you can thinkof a sphere as made up of four 2d circles projected into 3d space and hence the area is 4 times the area of a circle. This seems to me like a valid alternative proof?
I loved the video! It made the concept clear. If I watch this video 2 times to understand the problem, then, without this video, I would have understood the concept only after 200 times of reading the textbook!
@@stephyelle1 his "proof" was more of an experiment test by comparing the volume of a cylinder with the volume of a sphere plus a bicone. There's a numberphile video about this
Another elegant method is using the volume of the sphere to deduce it's surface area. The volume is 4 pi/3 r^3, curiously the derivative is 4 pi r² or the surface area. This is no coincidence. Take the function V(r) = 4 pi/ 3 r^3 and take the derivative. That is, (V(r+h)-V(r))/h as h goes to 0. Geometrically this represents the difference in volume between a sphere and a slightly bigger sphere. Then divide that by the difference in the radius, intuitively it's clear that you get better and better aproximiations of the surface if that difference get's smaller, so the derivative must be the exact surface area and there you have it. Very intuitive.
wow! With demonstrations like this, the schools would keep attention of students instead of them losing interest because they don't understand where the formulas come from ! Bravo!
Super high quality and very polished! Great for people who haven't learned calculus yet! For a (much shorter!!!) Proof using trigonometry and calculus, do a youtube search for "Proof of Surface Area of a Sphere" (Not my channel, just promoting another good video :)
brilliant explanation i think this explanation contain all procedure that we study from basic level....which is easily understandable but ....some teacher go directly to the formula and did not teach the basic concept ....i think every theorem should be taught like this way ......
That was cool. When you mentioned many little sides, I immediately jumped to the idea that limits were to be involved. (Technically they were, but is was phrased in a different way)
Theres a much simpler proof: To form a sphere, you must rotate a circle around its diameter. And, if you look, you can see that the surface area of the sphere is equal to the circumference of the shadow times the distance it was rotated. So we plug in: “SA=2πr*o” in which o is the distance the circle was rotated around. Now, if we look AGAIN, we can see that the distance it was rotated around was actually equal to the diameter. So next we plug in: SA= “2πr*2r”. Simplifying, we get “SA=4πr^2
Hey...I'm losing you. "To form a sphere, you must rotate a circle around its diameter." Okay, that makes sense. "The surface area of the sphere is equal to the circumference of the shadow times the distance it was rotated." Again, that makes sense - and the circumference of the shadow would be equal to the circumference of the circle. "So we plug in: “SA=2πr*o” in which o is the distance the circle was rotated around." Got you. "Now, if we look AGAIN, we can see that the distance it was rotated around was actually equal to the diameter." Wait a sec...why is the distance it was rotated around equal to the diameter? If I have a circle, and I rotate it by 180 degrees with a diameter of that circle as its axis, and let the points on the edge of the circle trace out a surface, points on the part furthest will have moved pi r, and points closer will have moved less...how did you get that the distance rotated around was equal to the diameter? Different points on different parts of the circumference of the circle rotate by different amounts.
if you wanna look at it using a different calculus approach then it's the derivative of the volume which makes sense if you think about how the surface area is pretty much the rate of change of the volume
You're right, however, as someone who's curious but not up to calculus yet, I really appreciated this proof. It was simple and only required a decent understanding of geometry and manipulating equations, making it more accessible to a far wider audience.
But how do you derive the volume? Btw what you stated isn't always true, for example in a cube the rate of change of the volume is only half of the surface area, cause increasing the side only affects one direction, which would be analougous to the derivative of the sphere volume with respect to d
I always get slightly confused when I think of it this stuff using derivatives. Like if you differentiate a circles area (pi r^2) then you get 2 Pi r - the circumference. Differentiate that and u get 2 Pi, the amount of radians in a circle. But what happens when you differentiate that? What’s that? And when you differentiate a spheres volume, you get the surface area, differentiate that and u get 8 Pi r - the circumference of a sphere??? It just leaves to many loose ends...
It takes an understanding that the differential surface area of a simply bounded function in three space is linked to the sum of the magnitude of the cross product along two bounded axis. The double integral with the differentials alone is the area, and the function defines the height at each mid point. You need to understand partial derivatives to understand what I'm talking about. Deriving the surface area of a sphere is straight forward, by what if you have a simply bounded elliptical hyperboloid with specified boundary parameters?
@@guitarttimman Take two coordinates (rcosx , rsinx) , people don't realize that if sinx is increased by r times then that new point , rsinx , is increased by r in the x direction. So before integrating all points on the circumference , rsinx must be multiplied by r again giving us r^2sinx. Then it's easy to integrate to get surface area take the definite integral of 2pir^2sinx over o-pi.
Thank you for your patient explanation, but I think there is still one flaw in the line of reasoning: since we derived the formula of the surface area of model = pi*AD*AE only when the number of sides of the inscribed polygon was 8, how could we use it for n is greater than 8?
That was amazing but I would have thought there would be a simpler explanation? Like using a hemisphere:- Surface area of a circular strip = pi * (r1+r2) * l As it goes to infinitesimal, r1 + r2 become the same, so 2r So 2 pi r * integral of all the ls would give the hemisphere. All the l's are straight lines along the radius, added up for the hemisphere gives r So 2 pi r^2 The multiply by 2 for the sphere: 4 pi r^2 Or is this insufficient proof?
If I wanted to just, say, take a circle, measuring only its circumference, then rotate that circle an increment of dθ, then basically keep rotating that circle dθ, summing up each circle's contributory radius until I went around 2π, i.e. integral from 0 to 2π of the circumference of a circle rotating about dθ, would that give me similar results? I find calculus gives somewhat more intuitive answers sometimes
The area formulae for the surface area of a cone and a frusturm are presented as though they are trivially obvious. If that's the case, then so is surface area of a sphere. But given that we are attempting to derive the latter, we should derive the former.
would have liked to see .5! on the graph and maybe points between the integers too. since 0! is 1 on the graph and sqrt(pi)/2 isn't one, what does the graph look like
Maybe it would be a good thing to tell, that the small formula r_1 + r_2 + r_3 = AE * AD / 2s works in a similar way for all polygons you choose. Otherwise the resulting area of the polygons might change while increasing the number of vertices.
What was the step that allowed for the approximation of the polygon's area to approach the surface area of a sphere? It went from 2-D to 3-D and I didn't see how
Wait a moment, why that extra step with r3 at 2:53, when it's equal to r1? We're talking about a sphere, so the left side and the right side are equally big. I like this video, but that one step is completely unnecessary. The cones on the left and right are identical, they are the same shape, the same length, even the same color (don't mind the last one), so why to label that line differently from the r1?
I posted some Calculus videos on my channel which is just a sample of what I know about the subject. I do an eloquent derivation using single integrals.
The overall approach is nice and your animation is very clear, but I don't think you formally generalised from the octagon you started with to the general 4n-gon that's required for the limit process to be valid. Or am I missing something?
Hey, I love your videos! They make everything so much clearer about math! I actually do not quite get proofs for the law of cosines, so I was hoping you could do a video on it. Thanks!
i had one doubt that the hypotenuse of a right angled triangle can never be the same as any other side hence AE can never be equal to AD so its a bit inaccurate but it surely does make sense when u round it off
Good job. But someone would want to know where the lateral area of a right circular frustum comes from (which is derived from the lateral area of a right circular cone).
I had figured it out on my own but wanted confirmation that I was correct. I was. Anyways, the point of this comment is that this video was beautifully illustrated and explained. Also, that math has many avenues by which one can reach the desired answer. What I did is I drew a sphere and drew two circles in it on the x, y and z-axis. Then I drew a separate diagram of one of the circles. I know that 2(pi)r or (pi)d were my circumference. I used (pi)d. I then imagined another diameter on the z-axis coming from the first circle. I then multiply (pi)d*d. I got(pi)d^2. I then converted d^2 to r. I got 4r^2. This gave me 4(pi)r^2.
BEAUTIFUL presentation! Clear. concise, organized, with good graphics and pacing.
Thumbs up and subbed!
how am i supposed to understand this stuff?
@@flower_girl4983 learn the basics first for example the properties of triangle and other basic shapes, then go for average (this video) and finally the difficult ones. That's how you can master the art of learning mathematics
@@sam-ui5lc ok sam thnks a lot
@@sam-ui5lc and limits too
One word: "Perfect!"
This presentation couldn't have been done better.
Thnks
An elegant method to derive the formula for the area of the surface of a sphere without using calculus.
***** Just intuitively, humans have been doing that in math long before calculus. One thing that is cool is that mathematicians were close to inventing calculus back in the ancient historical times. I forgot the details but I think I read it in a Lancelot Hogben book.
+Dan Kelly he used formulas for a cone and a frustum. How are you supposed to do it without calculus?
Kemerover He didn't use integrals did he? I only watched the surface area of a sphere part. Oh when you said cone I started to think about volumes sorry. Anyway not all formulas require calculus to derive them. Some formulas were derived about 1500 years before calculus. Some of the concepts that are used in calculus were developed about 2,000 years ago.
+Dan Kelly i agree on what he did.. i can understand this... but if he used calculus i coud not
Ivan Ereiz Yes he just used one of the concepts used in calculus but he didn't use the 'language' of calculus so this is a nice piece of work that everyone who has high school algebra can understand.
Thank you very much! This video is very helpful for students like me who have yet to learn calculus, but still want to understand what they're doing. I can usually come up with my own "proofs" for most formulas, but when it came to spheres I was completely lost. Now it makes sense to take a polygon with infinite sides, just as you do with the circular(ish?) part of cone. Thanks! Really hits home the beauty and creativity of math, especially for a subject that most people assume is dry with no room for creativity.
Check out the 3b1b episode too
I relate 100% to your comment!
The better way to spend your time is to learn calculus.
@@zachansen8293 yeah that would have been really easy during o levels man thanks bro should have just done that on top of studying only 2 months
Beautiful!
Raw simplicity & beauty of mathematics presented with clear & concise explanation and graphics. It doesn't get much better than this!
Thank you, thank you, thank you!
If you were expecting a simple answer...you were wrong.
That's why people are watching this video: the formula is so simple.
Hermes Mercury simple mind?
Simpler than integral calculus.
Well it wasn't THAT hard to understand :P
Hermes Mercury If the diameter is the same volume of the circumference, then it'd have a ratio of 4, am I wrong?
By far, the most elegant and unique derivation of the formula, without calculus, which makes it understandable to a larger number of students.
A mathematical elegance presented in clear and concise graphics and a truly immaculate approach.
It can't get any better. Thank you, on behalf of all the students who are not yet introduced to calculus!
Beautiful. Subbed instantly!
I totally agree.
Damn who came up with this -_- i understand but i would never think of something like this. Imagine you were in a time where all you had was a sphere in your hand and someone was able to think of this AMAZING
manipulating pre existing resources and bending it to fit to a new problem was my introduction to calculus,its not easy,but does not take a genius to do this, fascinating regardless
Fantastic - beautifully clear explanation.
Thnkx
Beautiful!
Just a random question emerged in my mind when I was solving physics problems turn out to has one of the most fascinating explanation about math that I've ever watch.
The video is clean and smooth, didn't expect this quality from a 2013 TH-cam Video.
Thank you so much!
Glad to hear you enjoyed it!
@@mathematicsonline Thanks for sharing but I just don't see how or why anyone would cone up with this at all? Especially since it's so convoluted and unintuitive. My idea for a proof is 4 pi r squared is 4 times the area of a circle so you can think of a sphere as having four "faces" like a box has four faces. So you can thinkof a sphere as made up of four 2d circles projected into 3d space and hence the area is 4 times the area of a circle. This seems to me like a valid alternative proof?
@@leif1075 It is an ancient proof by Archimedes, it gives us insight to early mathematics.
I can't believe this channel is not that popular omg it is precisely amazing
Thnks
Thnks
I loved the video! It made the concept clear. If I watch this video 2 times to understand the problem, then, without this video, I would have understood the concept only after 200 times of reading the textbook!
What do you use to edit the video? The animations are so clear and helpful.
Superb proof!
luca nina Archimedes proof.... 200 years before JC!
Thanks
@@stephyelle1 Amazing to think how Mathematicians used to derive this stuff back then when Maths wasn't this advanced
@@stephyelle1 his "proof" was more of an experiment test by comparing the volume of a cylinder with the volume of a sphere plus a bicone. There's a numberphile video about this
Thanks for this - I expected a very complicated explanation,but actually it all made sense. Great video.
interesting how the area of a circle is pi*r^2 but the (surface) area of a sphere is pi*d^2
That's an even simpler mnemonic tool.
thanks Ryan.
Ryan Bell thnx
The surface area of a sphere is 4*pi*r^2......
please explain someone
A= 4pi*r^2.... r= d/2....... so 4*pi*(d/2)^2 => 4*pi*(d^2/ 4).......4's cancel and all you have left is pi*d^2. Hope this helps.
This is the definition of a perfect video
It's crystal clear. I can use this way to enhance students understanding. The way I like that
Beautiful!
Simply...
Beautiful!
Thanks a lot for this simple explanation to the otherwise seemingly complicated problem.
Thank you!!!
Another elegant method is using the volume of the sphere to deduce it's surface area. The volume is 4 pi/3 r^3, curiously the derivative is 4 pi r² or the surface area. This is no coincidence. Take the function V(r) = 4 pi/ 3 r^3 and take the derivative. That is, (V(r+h)-V(r))/h as h goes to 0. Geometrically this represents the difference in volume between a sphere and a slightly bigger sphere. Then divide that by the difference in the radius, intuitively it's clear that you get better and better aproximiations of the surface if that difference get's smaller, so the derivative must be the exact surface area and there you have it. Very intuitive.
dekippiesip As UNBELIEVABLE as it looks, if U use derivatives, 4*Pi*(r^3)/3 turns into 4*Pi*(r^2)!
dekippiesip its*
dekippiesip approximation*
dekippiesip gets*
I think volume is derived using SA itself! By integrating SA for all r from 0 to R. So u can't use that.
Great. I have never seen a clear explanation like this!
complex concept, but brought forward in a simple and understandable manner. thanks a bunch man
The best explanation over youtube. Thank you very much.
wow! With demonstrations like this, the schools would keep attention of students instead of them losing interest because they don't understand where the formulas come from ! Bravo!
Very good comprehensive video. I always tend to take these formulas for granted.
Mathematics basics are explained very clearly . Great work nicely done. Thank you
wow. my jaw is on the floor. I loved how it all simplified so nicely in the end. Great video, btw!
To be honest with yall, this guy's explanation is excellent fr
Well done. Classic proof with great explanation and illustration.
Best explanation i ever seen on youtube.
Wow. This is like, proofs to the max. I've never seen such a complicated proof about spheres; great job!
Hi like im dad
Super high quality and very polished!
Great for people who haven't learned calculus yet!
For a (much shorter!!!) Proof using trigonometry and calculus, do a youtube search for
"Proof of Surface Area of a Sphere"
(Not my channel, just promoting another good video :)
Animations and explanations are best... thanks for making this types of videos.
Thank you. I was always wondering but never got such an explanation.
Thank you for your help.. 😊. Was looking forward for such theory and I guess I got what I wanted to see!
The best explanation I ever seen thanks buddy I'll be your subscriber forever
Appreciate it!
brilliant explanation i think this explanation contain all procedure that we study from basic level....which is easily understandable but ....some teacher go directly to the formula and did not teach the basic concept ....i think every theorem should be taught like this way ......
Nearly lost me for a moment but I'm very glad I stuck with it. November 2021. You just wouldn't believe what's been going on.
That was cool. When you mentioned many little sides, I immediately jumped to the idea that limits were to be involved. (Technically they were, but is was phrased in a different way)
That was just... BEAUTIFULLY done! Thank you!
mind blown ! i've never thought of this before it's a master piece
Math really does builds upon itself
One of the most helpful answers
Thanks. Excellent, logical and easy to follow.
Your videos are awesome and very informative and are on a different level from most explanations, Thank You.
Very intuitive. Maybe a comment that this derivation also applies to polygons with more than 8 sides, would be perfect.
Theres a much simpler proof:
To form a sphere, you must rotate a circle around its diameter. And, if you look, you can see that the surface area of the sphere is equal to the circumference of the shadow times the distance it was rotated. So we plug in: “SA=2πr*o” in which o is the distance the circle was rotated around. Now, if we look AGAIN, we can see that the distance it was rotated around was actually equal to the diameter. So next we plug in: SA= “2πr*2r”. Simplifying, we get “SA=4πr^2
Very nice bro
Hey...I'm losing you.
"To form a sphere, you must rotate a circle around its diameter." Okay, that makes sense.
"The surface area of the sphere is equal to the circumference of the shadow times the distance it was rotated." Again, that makes sense - and the circumference of the shadow would be equal to the circumference of the circle.
"So we plug in: “SA=2πr*o” in which o is the distance the circle was rotated around." Got you.
"Now, if we look AGAIN, we can see that the distance it was rotated around was actually equal to the diameter." Wait a sec...why is the distance it was rotated around equal to the diameter? If I have a circle, and I rotate it by 180 degrees with a diameter of that circle as its axis, and let the points on the edge of the circle trace out a surface, points on the part furthest will have moved pi r, and points closer will have moved less...how did you get that the distance rotated around was equal to the diameter? Different points on different parts of the circumference of the circle rotate by different amounts.
@@joshuaronisjr True, but the distance is constant, and it's equal to pi * r , as you move it "Half the sphere".
@@gligoradrian784 What distance is constant?
@@joshuaronisjr I mean, the 180* around which you rotate the circle, and also pi.
Awesome video. Animations were clear and helpful and the proof was simple and beautiful. Liked and Subbed!
2:51 r2... d2?
if you wanna look at it using a different calculus approach then it's the derivative of the volume which makes sense if you think about how the surface area is pretty much the rate of change of the volume
That's just "reducing" a simpler problem to a harder problem.
Nyx Avatar what is calculas
You're right, however, as someone who's curious but not up to calculus yet, I really appreciated this proof. It was simple and only required a decent understanding of geometry and manipulating equations, making it more accessible to a far wider audience.
But how do you derive the volume? Btw what you stated isn't always true, for example in a cube the rate of change of the volume is only half of the surface area, cause increasing the side only affects one direction, which would be analougous to the derivative of the sphere volume with respect to d
I always get slightly confused when I think of it this stuff using derivatives. Like if you differentiate a circles area (pi r^2) then you get 2 Pi r - the circumference. Differentiate that and u get 2 Pi, the amount of radians in a circle. But what happens when you differentiate that? What’s that? And when you differentiate a spheres volume, you get the surface area, differentiate that and u get 8 Pi r - the circumference of a sphere??? It just leaves to many loose ends...
U just made ur life harder bro good job
A double integral in polar coordinates works best!
What about surface Integrals?
+VivzStudioSs "Double integral"
VivzStudioSs Surface integral turns out to be a double integral
It takes an understanding that the differential surface area of a simply bounded function in three space is linked to the sum of the magnitude of the cross product along two bounded axis. The double integral with the differentials alone is the area, and the function defines the height at each mid point. You need to understand partial derivatives to understand what I'm talking about. Deriving the surface area of a sphere is straight forward, by what if you have a simply bounded elliptical hyperboloid with specified boundary parameters?
@@guitarttimman Take two coordinates (rcosx , rsinx) , people don't realize that if sinx is increased by r times then that new point , rsinx , is increased by r in the x direction. So before integrating all points on the circumference , rsinx must be multiplied by r again giving us r^2sinx. Then it's easy to integrate to get surface area take the definite integral of 2pir^2sinx over o-pi.
Simply beautiful, great video!
this was a very elegant and simple way to solve it, thank you!
Thank you for your patient explanation, but I think there is still one flaw in the line of reasoning: since we derived the formula of the surface area of model = pi*AD*AE only when the number of sides of the inscribed polygon was 8, how could we use it for n is greater than 8?
the best thing is when you can understand, that's proportionate by a good explanation, thank you. Muito bom, pena não haver canais assim em português.
Perfect ❤👏
Greetings to you from Egypt !!
Beautiful explanation!!
AMAZING. MAN OF THE PEOPLE RIGHT HERE.
You have made me do my homework.
Thank you very very very ....much.
That was amazing but I would have thought there would be a simpler explanation?
Like using a hemisphere:-
Surface area of a circular strip = pi * (r1+r2) * l
As it goes to infinitesimal, r1 + r2 become the same, so 2r
So 2 pi r * integral of all the ls would give the hemisphere.
All the l's are straight lines along the radius, added up for the hemisphere gives r
So 2 pi r^2
The multiply by 2 for the sphere: 4 pi r^2
Or is this insufficient proof?
wow what an awesome explanation 😊
If I wanted to just, say, take a circle, measuring only its circumference, then rotate that circle an increment of dθ, then basically keep rotating that circle dθ, summing up each circle's contributory radius until I went around 2π, i.e. integral from 0 to 2π of the circumference of a circle rotating about dθ, would that give me similar results?
I find calculus gives somewhat more intuitive answers sometimes
The area formulae for the surface area of a cone and a frusturm are presented as though they are trivially obvious. If that's the case, then so is surface area of a sphere. But given that we are attempting to derive the latter, we should derive the former.
We can prove it by integrals too. And I think it's better! But your proof is pretty good too!
Great explanation sir 😊😊
Who was hoping he said R2D2?? Please tell me I wasnt the only one
Hahahaahahaha
C^3*PO
Shynn Sup wasn't*
Any way to do this using shapes that are simpler than a sphere?
would have liked to see .5! on the graph and maybe points between the integers too. since 0! is 1 on the graph and sqrt(pi)/2 isn't one, what does the graph look like
can you do the surface area of the cone and frustum? I'm trying to understand this down to its beginnings
how do you make your vids? blender?
Maybe it would be a good thing to tell, that the small formula r_1 + r_2 + r_3 = AE * AD / 2s works in a similar way for all polygons you choose. Otherwise the resulting area of the polygons might change while increasing the number of vertices.
Indeed. The formulas are probably true when there are more than 2 frustrums, but that needs to be demonstrated, which it wasn't.
can't you keep a rectangle inside the circle for easy calculation of area
Wonderful explanation
Very elegant solution. Thanks for posting!
Nicely explained.
Conceptual answer. Good explanation
Amazing! I had no idea it was this complex!
How are the surface areas of the cone and frustum derived?
Beautiful explanation.
Thank you so much for this wonderful presentation.....
awesome graphics--what program are you animating with?
What was the step that allowed for the approximation of the polygon's area to approach the surface area of a sphere? It went from 2-D to 3-D and I didn't see how
Wait a moment, why that extra step with r3 at 2:53, when it's equal to r1? We're talking about a sphere, so the left side and the right side are equally big. I like this video, but that one step is completely unnecessary. The cones on the left and right are identical, they are the same shape, the same length, even the same color (don't mind the last one), so why to label that line differently from the r1?
I posted some Calculus videos on my channel which is just a sample of what I know about the subject. I do an eloquent derivation using single integrals.
BRO YOU ARE HEAVENLY!
BEAUTIFUL JUST BEAUTIFUL
The overall approach is nice and your animation is very clear, but I don't think you formally generalised from the octagon you started with to the general 4n-gon that's required for the limit process to be valid. Or am I missing something?
Hey, I love your videos! They make everything so much clearer about math! I actually do not quite get proofs for the law of cosines, so I was hoping you could do a video on it. Thanks!
Can you please explain why the triangle ADE is similar to other triangles.
Thnx a lot!!!
It really helped me out in my seminar.
U da BEST!!!!
this derivation was shocking fr me
well done 👍
I love it, Excellent explanation well done man! 👏👍
Just excellent
Please also make a video on formula of (A3-B3)=
i had one doubt that the hypotenuse of a right angled triangle can never be the same as any other side hence AE can never be equal to AD so its a bit inaccurate but it surely does make sense when u round it off
Good job. But someone would want to know where the lateral area of a right circular frustum comes from (which is derived from the lateral area of a right circular cone).
I had figured it out on my own but wanted confirmation that I was correct. I was. Anyways, the point of this comment is that this video was beautifully illustrated and explained. Also, that math has many avenues by which one can reach the desired answer. What I did is I drew a sphere and drew two circles in it on the x, y and z-axis. Then I drew a separate diagram of one of the circles. I know that 2(pi)r or (pi)d were my circumference. I used (pi)d. I then imagined another diameter on the z-axis coming from the first circle. I then multiply (pi)d*d. I got(pi)d^2. I then converted d^2 to r. I got 4r^2. This gave me 4(pi)r^2.
This is amazing.. Crystal clear!
this is the best proof i've seen.
Nice presentation.
I like your explanations