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.

[@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?

[@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

[@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

[@lifeisascam]

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.

[@mujtabaalam5907]

Check out the 3b1b episode too

[@IsaacAsimov1992]

I relate 100% to your comment!

[@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

[@user-hl5xh5jr4d]

It's crystal clear. I can use this way to enhance students understanding. The way I like that

[@rarebucko]

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

[@Singh-be2qn]

Very nice bro

[@joshuaronisjr]

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.

[@gligoradrian784]

@@joshuaronisjr True, but the distance is constant, and it's equal to pi * r , as you move it "Half the sphere".

[@joshuaronisjr]

@@gligoradrian784 What distance is constant?

[@gligoradrian784]

@@joshuaronisjr I mean, the 180* around which you rotate the circle, and also pi.

[@markhatton6449]

Fantastic - beautifully clear explanation.

[@mikasaackerman3946]

Thnkx

[@Alex_science]

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

[@portalsrule1239]

wow. my jaw is on the floor. I loved how it all simplified so nicely in the end. Great video, btw!

[@ketofitforlife2917]

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

[@leenagupta6586]

Thank you for your help.. 😊. Was looking forward for such theory and I guess I got what I wanted to see!

[@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.

[@shampadutta7322]

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!

[@IsaacAsimov1992]

I totally agree.

[@tearchi]

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

[@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!

[@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!

[@guhaonkar]

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

[@acpf4f]

Thanks. Excellent, logical and easy to follow.

[@skrd37]

The best explanation over youtube. Thank you very much.

[@jackmack1061]

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.

[@joeywarren]

Well done. Classic proof with great explanation and illustration.

[@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.

[@pauldifolco5736]

Awesome video. Animations were clear and helpful and the proof was simple and beautiful. Liked and Subbed!

[@shotaaizawa1888]

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

[@SoumilSahu]

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

[@user-rs8965grt]

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

[@gavincraddock5772]

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

[@banajadandasena4142]

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

[@josephprashanthbritto8349]

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

[@math2693]

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

[@mikasaackerman3946]

Thnks

[@mikasaackerman3946]

Thnks

[@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)

[@eeltauy]

Amazing! I had no idea it was this complex!

[@MegaJayPower]

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

[@chessandmathguy]

Very elegant solution. Thanks for posting!

[@sylvainpilon9496]

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!

[@DeathScakez]

mind blown ! i've never thought of this before it's a master piece

[@Junnnn___]

To be honest with yall, this guy's explanation is excellent fr

[@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.

[@almamashkulli5349]

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

[@joaopedrob.rodrigues4945]

Simply beautiful, great video!

[@johnaugsburger6192]

Very well done, I love this stuff.

[@Geometria101]

A great and clear explanation. Thank you.

[@travisbaskerfield]

Beautiful explanation.

[@Pr0Puffin]

This is a brilliant rendition of the the solution which does not involve calculus. It was quite useful. Great work guys.

[@JustAgreenBoy6969]

Actually It has limits in this proof but that's just too ezzy

[@Pr0Puffin]

@@JustAgreenBoy6969 thanks for replying; I had forgotten about this video and comment for ages. I wrote that comment when I was barely 13; now I’m studying engineering in college - time flies.

[@JustAgreenBoy6969]

@ProPuffin cool bro I'm 14. Atb for your engineering

[@AdolfPA]

Classy & clear explanation. Thanks

[@tomasmalm660]

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

[@MsLouloulepou]

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

[@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.

[@mugundansridhar3835]

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

[@akshitasiddhapura4626]

One of the most helpful answers

[@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

[@agarykane2127]

Beautiful explanation,thank you very much!

[@brunoghilardi5976]

Just excellent

[@Inu1907]

This is absolutely beautiful!

[@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?

[@mauk2009]

Best explanation i ever seen on youtube.

[@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!!

[@joy2000cyber]

Very intuitive. Maybe a comment that this derivation also applies to polygons with more than 8 sides, would be perfect.

[@efecanacar9875]

This is amazing.. Crystal clear!

[@MarkDLongo]

Love your videos, keep it up!

[@DrYacineKoucha]

Beautiful explanation!!

[@raheena9881]

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

[@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...

[@dsy9578]

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

[@mathematicsonline]

Appreciate it!

[@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

[@skatelife59]

that is a beautiful derivation...

[@wasfiiwasfi]

i love this kind of demonstration !!

[@jacobmerrill1647]

mathematicsonline, I've loved your videos. Quick question: near the end of this video, where you use the animation to simulate increase of the inscribed polyhedron's sides, must line AD really move/change? Its moving to ultimately conform to the diameter didn't seem right to me, even though it made sense by animation.

[@Ghostalitta]

I love it, Excellent explanation well done man! 👏👍

[@B20C0]

This is actually amazing.

[@nitishmohanty8726]

You have made me do my homework. Thank you very very very ....much.

[@monikagoyal7227]

I hadn't thought it's so complex

[@kannusingh7003]

I like how the color of the 3d model become black at infinity which is extremely true

[@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*

[@tapasbanerjee7936]

Nicely explained.

[@banajadandasena4142]

Conceptual answer. Good explanation

[@ezrapotter4631]

From a calculus standpoint, the surface area is the derivative of the volume, 4/3pi(r^3)

[@atanasiok]

Excellent explanation!!!

[@AhmedRamadan-mc4dt]

Perfect ❤👏 Greetings to you from Egypt !!

[@anuragchavan7900]

Wonderful explanation

[@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 ......

[@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.

[@sidaliu8989]

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?

[@morgengabe1]

Any way to do this using shapes that are simpler than a sphere?

[@abdullahnoman8618]

I like your explanations

[@Name-ps9fx]

Awesome thought process! Someone was much smart.

[@HecticHector]

U just made ur life harder bro good job

[@ayushghosh3912]

BRO YOU ARE HEAVENLY! BEAUTIFUL JUST BEAUTIFUL

[@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

[@sanaislam90]

this derivation was shocking fr me well done 👍

[@mohanbuvan]

Nice presentation.

[@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?

[@saritadigrawal7823]

wow what an awesome explanation 😊

[@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

[@chapalex1872]

We can prove it by integrals too. And I think it's better! But your proof is pretty good too!

[@NovaWarrior77]

AMAZING. MAN OF THE PEOPLE RIGHT HERE.