Understanding the Surface Area of a Sphere Formula

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Deriving the formula.
Proof and explanation that Surface Area of a Sphere is equal to 4πr^2 using geometry and algebra.
The surface area of a sphere is the area occupied by the surface of the sphere.

[@dannygjk]

An elegant method to derive the formula for the area of the surface of a sphere without using calculus.

[@dannygjk]

***** 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]

+Dan Kelly he used formulas for a cone and a frustum. How are you supposed to do it without calculus?

[@dannygjk]

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.

[@ivanereiz1533]

+Dan Kelly i agree on what he did.. i can understand this... but if he used calculus i coud not

[@dannygjk]

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.

[@Westkane11]

One word: "Perfect!" This presentation couldn't have been done better.

[@mikasaackerman3946]

Thnks

[@learnerlearns]

BEAUTIFUL presentation! Clear. concise, organized, with good graphics and pacing. Thumbs up and subbed!

[@flower_girl4983]

how am i supposed to understand this stuff?

[@sam-ui5lc]

@@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

[@mikasaackerman3946]

@@sam-ui5lc ok sam thnks a lot

[@JustAgreenBoy6969]

@@sam-ui5lc and limits too

[@hermesmercury]

If you were expecting a simple answer...you were wrong.

[@levi4328]

That's why people are watching this video: the formula is so simple.

[@stephyelle1]

Hermes Mercury simple mind?

[@maacpiash]

Simpler than integral calculus.

[@TheZMDX]

Well it wasn't THAT hard to understand :P

[@landlord112]

Hermes Mercury If the diameter is the same volume of the circumference, then it'd have a ratio of 4, am I wrong?

[@anthonygopeesingh7645]

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

[@davidsica8996]

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!

[@lucanina8221]

What do you use to edit the video? The animations are so clear and helpful. Superb proof!

[@stephyelle1]

luca nina Archimedes proof.... 200 years before JC!

[@mikasaackerman3946]

Thanks

[@mustafahassan3584]

@@stephyelle1 Amazing to think how Mathematicians used to derive this stuff back then when Maths wasn't this advanced

[@mujtabaalam5907]

@@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

[@dekippiesip]

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.

[@tonybidwell2582]

dekippiesip As UNBELIEVABLE as it looks, if U use derivatives, 4*Pi*(r^3)/3 turns into 4*Pi*(r^2)!

[@JorgetePanete]

dekippiesip its*

[@JorgetePanete]

dekippiesip approximation*

[@JorgetePanete]

dekippiesip gets*

[@kartikraj1779]

I think volume is derived using SA itself! By integrating SA for all r from 0 to R. So u can't use that.

[@markhatton6449]

Fantastic - beautifully clear explanation.

[@mikasaackerman3946]

Thnkx

[@sushantnair2584]

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!

[@curs3d975]

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 KZbin Video. Thank you so much!

[@mathematicsonline]

Glad to hear you enjoyed it!

[@leif1075]

@@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?

[@mathematicsonline]

@@leif1075 It is an ancient proof by Archimedes, it gives us insight to early mathematics.

[@danmarino900]

interesting how the area of a circle is pi*r^2 but the (surface) area of a sphere is pi*d^2

[@commentercommenting328]

That's an even simpler mnemonic tool.

[@acpf4f]

thanks Ryan.

[@purushottammotwani3082]

Ryan Bell thnx

[@Bluedragon2513]

The surface area of a sphere is 4*pi*r^2...... please explain someone

[@BouncingHope]

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.

[@math2693]

I can't believe this channel is not that popular omg it is precisely amazing

[@mikasaackerman3946]

Thnks

[@mikasaackerman3946]

Thnks

[@gavincraddock5772]

Thanks for this - I expected a very complicated explanation,but actually it all made sense. Great video.

[@Alex_science]

Great. I have never seen a clear explanation like this!

[@tylerpruitt9572]

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

[@swn32]

That's just "reducing" a simpler problem to a harder problem.

[@QwertQwert-qo3le]

Nyx Avatar what is calculas

[@mr.moodle8836]

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.

[@tiscojack]

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

[@connoribbotson1337]

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...

[@guhaonkar]

Beautiful! Simply... Beautiful! Thanks a lot for this simple explanation to the otherwise seemingly complicated problem. Thank you!!!

[@shotaaizawa1888]

complex concept, but brought forward in a simple and understandable manner. thanks a bunch man

[@hamiltondepaula]

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.

[@mathmaticalproblemandsolution]

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 ......

[@Elseano14]

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)

[@lyrimetacurl0]

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?

[@noelb684]

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.

[@josephprashanthbritto8349]

Mathematics basics are explained very clearly . Great work nicely done. Thank you

[@JohnDixon]

Wow. This is like, proofs to the max. I've never seen such a complicated proof about spheres; great job!

[@odysseytkl7261]

Hi like im dad

[@alxjones]

This video should be called "How to derive the surface area of a sphere (assuming you somehow know the surface area of a cone and a frustum)". If you're going to approximate the sphere with cones and frustums, why not approximate those surfaces with triangles and trapezoids? Deriving the area of those objects is actually pretty easy, so you only need to derive those simple polygonal areas and you can derive this fact. This is more useful as a derivation than assuming knowlodge of the surface area for some uncommon solids.

[@MegaJayPower]

Very good comprehensive video. I always tend to take these formulas for granted.

[@user-rs8965grt]

Thank you. I was always wondering but never got such an explanation.

[@ketofitforlife2917]

That was just... BEAUTIFULLY done! Thank you!

[@TheHolyReality]

Everything in our pathetic existence and universe is just approximation . We cant ever calculate anything to exact value. Its all abstract. Nothing has any meaning. I wish i was never born or at least that i never learned things in my fucking life. Knowledge is only misery and sadness. Thank you for this excellent video.

[@WiperTF2]

Damn.

[@Kaldor-Draigo-h6q]

What?

[@NKPyo]

I lost you at 'Holy'

[@PhysicsOfParkour]

HAHA It's an Approximation because of TIME, things are constantly moving so even if u calculate things considering atoms; that state will instantly change because u cant stop time XD

[@spoderman15]

well somebody hasn't taken calculus. but still dude. youtube isn't you're psychiatrist

[@acpf4f]

Thanks. Excellent, logical and easy to follow.

[@skrd37]

The best explanation over youtube. Thank you very much.

[@banajadandasena4142]

Animations and explanations are best... thanks for making this types of videos.

[@smacksille1951]

An explanation that is easy for students to grasp is the physical size relationship of the inscribed circle to the square or cube. So this is for students looking to work out how to calculate the volume or area of a sphere. A ratio of a square to inscribed circle is approx. ¾ or 0.75 or π/no edges = 4 … So, if your circle is 2cm diameter, then the square is 2cm wide x 2cm high and its perimeter is 2cm * 4 edges (8cm). The corresponding circle that inscribes the box is therefore about 6cm or (8cm * (π / 4)). The area of the box is 4 cm2 so the area of the circle is about 3 cm2 or (4 cm2 * (π / 4)). ratio = π / 4 ratio * perimeter(square) = circumference(circle) cm ratio * area(square) = area(circle) cm2 Similarly constructed ratio of a cube to inscribed sphere is approx. ½ or 0.5 or π /no faces = 6 … So, if your sphere is 2cm diameter, then the box is 2cm wide x 2cm high x 2cm deep and its surface area is 4 cm2 * 6 faces (24 cm2). The area of the corresponding sphere that inscribes the box is therefore about 12 cm2 or (24 cm2 * (π / 6)). The volume of the box is 16 cm3 so the volume of the sphere is about 8 cm3 or (16 cm3 * (π / 6)). ratio = π / 6 ratio * area(cube) = area(sphere) cm2 ratio * volume(cube) = volume(sphere) cm3 I realise that this may be obvious to everyone here, but the reason I mention this is just that ratios seem to be a much simpler way for students to grasp the concept that an inscribed circle has a linear relationship to the area and perimeter of the square that bounds it … as does a inscribed spheres area and volume to the bounding box. Students easily grasp the volume of a box, by counting blocks, and knowing the relationship of a corresponding sphere is a fixed ratio, allows them to explore how that ratio was derived.

[@smacksille1951]

To determine any ratio from a regular polygon for a circle inscribed within the following formula can be applied to both perimiter and area: n = number of sides eqn = pi/n.tan(pi/n) tri = pi/3.tan(pi/3) sq = pi/4.tan(pi/4) hx = pi/6.tan(pi/6) oc = pi/8.tan(pi/8)

[@HecticHector]

U just made ur life harder bro good job

[@dsy9578]

The best explanation I ever seen thanks buddy I'll be your subscriber forever

[@mathematicsonline]

Appreciate it!

[@shynnsup8383]

Who was hoping he said R2D2?? Please tell me I wasnt the only one

[@cosmopolitan4598]

Hahahaahahaha

[@connorcriss]

C^3*PO

[@JorgetePanete]

Shynn Sup wasn't*

[@chloroformed8692]

Truly one of the proofs of all time

[@sriramhathwar9180]

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!

[@tearchi]

Your videos are awesome and very informative and are on a different level from most explanations, Thank You.

[@geckchanhong4513]

It is a beautiful proof. I enjoy reading it. Just one comment on how the prove can be generalize to (r1 + r2 + ... rn) = AE * AD / (2s) by specifying "diagonal" lines are between two consecutive vertical lines and the triangle form are similar. I do not see how the extension is achieved when I first read the proof. May a diagram of more triangle with ... between is shown. Once again, thank for the excellent presentation. I love it! Also, one observation, the angles are the same because there subtend the arc length.

[@mathematicsonline]

Thank you for your comment!

[@anzatzi]

awesome graphics--what program are you animating with?

[@SoumilSahu]

this was a very elegant and simple way to solve it, thank you!

[@EDUARDO12348]

I was looking at the formula for a sphere the other day in a math book expecting myself to derive this formula in my head, clearly, my brain would have exploded if I had really tried.

[@robkb4559]

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?

[@haroonrasheed3844]

Its a beautiful proof. But I have one question. The realtionship you derived in the form of r1 + r2 + r3 is only applicable to an octagon. And so is the area formula you wrote down for the frustums and cones. So when you increase the sides of the polygon, shouldn't the area formula and the r1+r2+r3 relationship become wrong? Why is that allowed?

[@anilkumarjain214]

How did you made these 3d objects? I was only staring at 3d objects. That was amazing. 🙏🙏🙏

[@eeltauy]

Amazing! I had no idea it was this complex!

[@mewsis14]

In Calculus we DERIVED the surface area as well as the volume of a sphere. 4/3 Pi (r)3 was derived.

[@AlchemistOfNirnroot]

Archimedes didn't have integral calculus.

[@DulksVenee]

I don't want to boast about this, but we had this assignment in the exam to prove 4(pi)r^2 to be a sphere's area, and I got the max score. Of course we were given the required formulae. I assume that's because they assumed that the genius who discovered this had his notes to help him.

[@divyanshusah8311]

Please also make a video on formula of (A3-B3)=

[@tapasbanerjee7936]

Nicely explained.

[@bobvonbuelow9983]

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

[@anniezhou8930]

OMG thanks so much for the vid!! help me alot!! do u mind making a video explaining how to calculate a specific portion of a sphere? because i'm doing my math extended essay on surface integrals and i find it hard to understand. Thanks so much!!

[@patrickfeng5066]

When you started aproxomating the figure to infinity, wouldn't all of your work with the similar triangles been rendered useless? All the work assumed that the figure was 8 sided.

[@cheongziyong8871]

They work for any polygon with an even number of sides

[@DrYacineKoucha]

Beautiful explanation!!

[@AhmedRamadan-mc4dt]

Perfect ❤👏 Greetings to you from Egypt !!

[@918019506206]

Thank you. It's a wonderful and a lot benefiting video. Please, can you also make videos to explain surface areas of cube and cuboid?

[@keiichiiownsu12]

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

[@WilliamMcCormickJr]

Diameter times Diameter times pi equals the area of a sphere. The volume of a sphere is Diameter times Diameter times Diameter times pi, divided by six. Sincerely, William McCormick

[@banajadandasena4142]

Conceptual answer. Good explanation

[@WilliamMcCormickJr]

The square of the length of one side of a cube, times six gives you the surface area of a cube. A sphere that exactly fits inside that cube, having the same diameter as the length of any side of that cube. Will have an area equal to pi times, the diameter times diameter, of that sphere. The volume of a cube is length times width times height. The volume of a sphere is diameter, times diameter times diameter, times pi, divided by six. So the volume of a cube times pi divided by six will give you the volume of a sphere with a diameter equal to one side of that cube. Sincerely, William McCormick

[@tryhardofdoom7682]

Why can you approximate it to infinity :( Yes it does make sense intuitively when looking at the last equation, but you derived it from the sum of 4 surface areas. This is driving me so crazy

[@colemanliam1]

calculus

[@alexharkler]

TryHard ofDoom Regardless of how many surface areas you have, the equation remains the same. You just end up cancelling out more "r"s.

[@circular17]

That was indeed missing in the explanations

[@tryhardofdoom7682]

Well, I didn't think people would take this comment seriously after 8 months... Anyway, I just wanted to say that I take the comment above back, it is really easy to understand why you can simply increase the number of sides of that polygon to infinity (atleast as long as the number of sides is even). If you don't just listen to what he says but take some time to think beyond what he says, you will find the answer :D

[@circular17]

No offence. There is just some room for improvement in the video. Everything is explained in great details except for this transition.

[@weirdshamanwizzard3156]

The guy who cane up with this clearly had a love for geometry

[@curtisnotestine3134]

How are the surface areas of the cone and frustum derived?

[@sreedevi1030]

Super explanation Thank you Not only with octogon We can also try it with hexagon

[@joaopedrob.rodrigues4945]

Simply beautiful, great video!

[@SocratesAlexander]

how do you animate? it seems very time consuming if you do it without any automation.

[@pivo6499]

that was fucking beautiful

[@nat4466]

lol

[@SkillslliK]

Holy hell, this was masterpiece.

[@mikasaackerman3946]

Thnks

[@kshitizpoudel-jp7cc]

can't you keep a rectangle inside the circle for easy calculation of area

[@najibqunoo7232]

9:05 you mean that AE and AD are both equal to the diameter of the sphere ; so here you will have it like this SA=pi*AE*AD SA=pi*d*d SA=pi*(d^2)

[@tnguyko]

holy crap. my brain just exploded watching this.

[@qwerty11111122]

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

[@weeklyshoe2201]

Watching this in math class :)

[@sanaislam90]

this derivation was shocking fr me well done 👍

[@ayushghosh3912]

BRO YOU ARE HEAVENLY! BEAUTIFUL JUST BEAUTIFUL

[@mugundansridhar3835]

Thnx a lot!!! It really helped me out in my seminar. U da BEST!!!!

[@raheena9881]

Thank you so much for this wonderful presentation.....

[@TroyaE117]

A simpler method is as follows... 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

[@origonalname119]

+TroyaE117 The last thing I'd call that is 'simple'.

[@TroyaE117]

+origonalname119 I never said it was "simple". I said it was "simpler".

[@TroyaE117]

I forgot to differentiate the volume with respect to r to get the surface area....d/dr of (4/3)*pi*r^3 = 4*pi*r^2Sorry.

[@stephyelle1]

Everything is Clear. Thanks, we can travel With Archimedes before calculus came witz Leibniz/Newton. Graphics are great support and please, don't listen to those who want music on your intro. Ridiculous, Event if I okay music for livin'

[@racheb93]

This formula can also quite easily be derived by taking the volume of a sphere and subtracting the volume of a slightly smaller sphere. Then you let the radii of those two spheres approach each other and get the same result.

[@Untoldanimations]

but don't you need the surface area of a sphere first to derive the volume formula?

[@pallabideka6729]

Ping Pong Cup Shots absolutely right

[@saritadigrawal7823]

wow what an awesome explanation 😊

[@ShehabEllithy]

You are great. I have longed to understand the formula

[@Everest314]

So if we already know how circles and rotational bodies work at the beginning, why don't we just construct the rotational body of a circle?

[@johnno4127]

This does not show the step between a 4 frustum approximation and an 'n' frustum approximation of a sphere. (Counting cones as frustums with a top radius of 0.)

[@ritikmehta7964]

Great work

[@Geometria101]

A great and clear explanation. Thank you.

[@MsLouloulepou]

Thanks so much for your sharing. It’s crystal clear.

[@highguardian13]

4:58 I bet r2d2 is the only reason you named the sides r and d in the first place

[@tomasmalm660]

Thanks, very nice demonstration! What tools do you use to create the math pics?

[@asal589]

I get it but I don't get it, logically, my brain is struggling between two concepts, when he says 4*pi*r^2, i think of just (2*pi*r)*(2*r) and when I think of that, I think of a hollow cylinder with height 2r, and that has no bases, and given that the diameter of the sphere and the height of the supposed cylinder are both 2r, it just seems logically contradictory to grasp that such a formula would be to find SA when in fact it isn't. I mean, a cylinder, the area in a sense is constant per unit of r/n (n being any arbitrary number) as you go cross the height, while in a sphere, the area is less at first, then peaks up to the center then decreases again per unit of r/n, i mean it just doesn't make sense to me. I hope people know what I'm saying here and if do, I really hope they can help clear this logical knot in my head

[@phoenixamaranth]

Take your cylinder and cut the ends so they can fold into a point on each end, then smush your cylinder so it bends and bulges outward at 3 points, once in the middle and twice equal distance from where your cuts end. You now have the same shape with the same surface area as his. Your height r of the cylinder was reduced in this process. The value of r would be different as the shape is morphed over the curve. Proportionality is maintained as the radius in the center of the cylinder expanded while the radius of the height diminished. Does that make sense?

[@larryderbin3538]

since radius is a measure from a center foci to the surface, if you have 2r x 2r you cannot make a cylinder shape... you fundamentally need to understand that although a cylinder may have a HEIGHT of 2r that means r is some arbitrary distance, while in this formula r is specifically only radius, the surface area of a cylinder is 2pi*h because the radius of a cylinder to the absolute center changes at every single point over the entire surface area

[@kawaiiirish6246]

I need to go now.. my head is hurting again btw nice video

[@jaimeduncan6167]

Very good derivation

[@mohanbuvan]

Nice presentation.

[@Name-ps9fx]

Awesome thought process! Someone was much smart.

[@jamesrocket5616]

Me: *Sees elegant* *Headmaster Henry Henderson enters the chat*

[@agarykane2127]

Beautiful explanation,thank you very much!

[@almamashkulli5349]

Thank you for this amazing video!You helped me a lot😊

[@GregLoutsenko]

how do you make your vids? blender?