But why is a sphere's surface area four times its shadow? 

Can we just take a moment to admire the quality of all animations in these videos ? It's just insane

7:53 "sometimes easy to forget why we're doing this".. its amazing how you pay attention to where one might loose track, after having mastered the topic.

The production quality of these videos never ceases to amaze me: The fluiditiy of the animations seamlessly demonstrating the ideas as they are being narrated. The impeccable pacing in the script that dives into the real unexpectedness and wonder of math. The passion and care that are woven through it all. You really are today's Feynman to me! Thank you!

Brown: *explains* Blue: But why? Brown: *angry noises*

Normal people : Why? 3Blue1Brown : But why?

him: unwraps a circle into a triangle me: you CaN dO tHaT?

This guy knows how to explain every detail and knows exactly what questions will be asked and immediately answers them. This guy is truly amazing!

I grew up in the "primitive era", learning math was murder. We've come a long way in the last 60+ years.

An absolutely brilliant video! I don't understand how the visualisations get better and better through each video, simply superb. I particularly enjoyed the alternating flashing technique to emphasise complicated parts of the video. You are the Shakespeare of Maths please never stop I am sure you will inspire some great minds in the future.

But why on earth maths isn't taught like this?

I remember trying to prove this in highschool. It seemed impossible given the knowledge of a child but I wonder if I had a teacher like you that time, it would have been an enlightened day of my lifetime. Thanks for these elegant proofs ❤️

I love this channel. It answers the WHY, the question no one ever seems to answer. All my math teachers have just re-explained the formulas, without ever saying how it actually works.

See most people find maths stressful and anxiety inducing... but somehow, this man has made it relaxing and beautiful.

It helps a lot if you remember that Oscar Had A Hold On Arthur. Answers below the fold: Q1: Let's call the radius of the ring d. We have a right triangle with an angle of theta and a hypotenuse of R. In this case, d is opposite from theta. Using the above mnemonic, we can remember that O/H = sin(theta), ergo d = R sin(theta). The circumference of a circle is 2 pi R, so the inner circumference of the ring is *2 pi R sin(theta).* Thus the ring's area is approximately *2 pi R^2 sin(theta) d(theta).* Q2: The good news is that our inner radius d is the same as it was for the ring on the sphere, ergo the inner circumference will also be the same: 2 pi R sin(theta). What we need to figure out is the thickness of the ring's shadow. By drawing another right triangle where the hypotenuse is the thickness of the ring, R d(theta), we can see that the thickness of the shadow is adjacent to theta in our new triangle. Using the mnemonic above, we can see that A/H = cos(theta), ergo the thickness of the shadow = R cos(theta) d(theta). To finish off, we multiply these two to get an area of *2 pi R^2 sin(theta) cos(theta) d(theta).* Q3: Using the identity that 2 sin(theta) cos(theta) = sin(2*theta) reveals that we can rewrite the area of the shadow as *pi R^2 sin(2*theta) d(theta).* This is the same as the area of the ring except that we've dropped the 2 from in front, signifying that we've cut the value in half, but we've also doubled the value of theta. This means that the shadow at a given value of theta has half the area of the ring at double that theta value. For example, the shadow at theta = 30 degrees has half the area of the ring at 60 degrees. Thus, as we go to the next shadow, we skip past one of the rings and jump two rings ahead instead of one. Q4: Partially answered above, but as we compare each shadow to a ring on the sphere, we have to skip every other ring, jumping two rings ahead for each shadow. The other half of this puzzle is to remember that we only generated shadows from one hemisphere rather than the entire sphere. Since we skip one ring for each shadow, that means we need to use all of the rings from the entire sphere (except for the ones we jump over), instead of just using the rings from one hemisphere. An easy way to see this is to think about the last shadow, at theta = 90 degrees, which corresponds to half the area of the ring at 180 degrees, which is the last ring on the sphere. Q5: The area of the shadows sums up to the area of a circle of radius R. However, each shadow is only _half_ the area of one of the rings, and only half of the rings have been accounted for. A half of a half is one quarter. Ergo, a circle of radius R only has half the area of one hemisphere of the sphere, which in turn only has half the area of the whole sphere, and thus the area of the circle is one quarter that of the entire sphere.

Grant, your math, narration and animation come together seamlessly like a Swiss watch. These videos are the very definition of 'Excellence'.

I am not interested in math. And yet, your videos are amazing, one of the few ones on RU-vid I watch fully. They are clear, calming, Super simple, a great way to spend time, always make sense and make simple things that looked like magic and wizardry back in school look like Kindergarten math. Wow. Thanks.

The most amazing thing is that Archimedes found the surface area of a sphere over a thousand years before calculus was even invented. While he didn't prove it to modern rigor, he can hardly be blamed for that.

A couple years ago, I was trying to estimate the amount of paint needed for an airliner. I based it on some simplifying assumption with regard to shapes and was surprised to find an equivalence between a hemisphere capped cylinder and open ended cylinder of the same overall length. Made me double check my math for the generalized case and indeed, (2 pi r)(l + 2r) describes both cases. So for my estimate paint usage, I just used the open ended cylinder with the length and radius for the fuselage and engines, plus a rough combination of triangles for the tail. (Wings, too, but their numbers stay separate since they use different products to prevent adhesion of ice)

Your videos are stunning in their simplicity, a perfect blend of math, computer, programming, and speech. Well done!

I've never seen such a powerful animation study in my life. Great job man.. keep going 💪

Your channel is just ABSOLUTELY AMAZING! I love your videos! Thank you for producing such nice Math content for the world. 👏🏻👏🏻👏🏻

3Blue1Brown: *Explaining mathematical concepts better than school ever could*

Man, I was just on a binge of shorts and this really feels like a great exit point for me into a longer format video ❤

The animation is spectacular and the explanation is so well done!

Congrats on such an amazing video, omg it's 3AM but here's how I did the exercise proof: 1. The circumference length of each ring is 2*pi*R*sin(theta), since the distance from the ring to the axis is R* sin(theta) (trigonometry). Hence, the area of a ring is 2*pi*R*sin(theta)*R*d_theta = 2*pi*R^2*sin(theta)*d_theta (1) 2. To calculate the area of a ring's shadow, I used some trig relations as well. In this case, the thickness of the ring shadow is R*d_theta*cos(theta). Therefore, the area of the ring shadow is 2*pi*R^2*sin(theta)*cos(theta)*d_theta (2) 3. Multiplying by both sides expression (2), we get 2*pi*R^2*2*sin(theta)*cos(theta)*d_theta I've put the number 2 right next to sin(theta)*cos(theta) to explicit the trigonometrical relation: 2*sin(theta)*cos(theta) = sin(2*theta) That specific angle is 2*theta. 4. So, this means that the area of the shadow of some ring with a corresponding angle of theta is equal to the area of a ring which has double of this angle. 5. Notice that, when theta ranges from zero to pi/2, we get to form all of the rings related to their shadows in the superior hemisphere. The total area of the shadow is the sum of all the thin ring shadows. Thus, that is equal to half the area of the hemisphere. That is, the area of the shadow (pi*R^2) is 1/4 of the surface area of the sphere. A(sphere) = 4*pi*R^2 Q.E.D.

Make a video on how 3 cones make a cylinder

Me at 3am: I don't need sleep, I need answers

THANK YOU. All the animations superbly reflect the thought process required to understand

I love 0:40 when he tried to cover the surface of the sphere with circles

I'm so smart I did the exercice all by thought without a piece of paper And I also got it all wrong

this channel is so genuine and passionate about maths!! really love it

Straightforward, informative, love your vids!

I'm watching this video after a year and it makes much more sense to me than ever. I remember when I first learnt about the surface area of sphere, it itched me and I searched RU-vid for that but it didn't made that much sense to me but now, I'm satisfied. Thanks 3Blue1Brown!

The math class is extended because the 3rd pi is still asking “why”. Whole class hates that pi.

Your explainations and animations break everything down into small logical understandable building blocks. Thank you so much!

I enthusiastically agree that demonstrations of finite sums of approximating at ever increasing fidelity the target result is a sometimes essential intellectual demonstration to get students over the "hump" of accepting limit theorems.

Please write a book on the beauty of mathematics!!!

*ADMIT* *IT* The *_Beauty_* *_of_* *_mathematics_* is the most satisfying thing ever..

The proof to the surface area explained via successive questions felt like a beautiful journey to uncovering a beautiful fact. It kinda gave me a feeling that i could've known it all along. Really fun, appreciate it :D

The animation and ideas are legit...!!! Thank you for making these gorgeous videos!!!

Huh. The "unwrapping 4 circles into triangles and fitting them into a cylinder" method is how I have been explaining this to students. I thought I came up with it. Bummer...

I really love how he explains WHY certain concepts work the way they do, instead of what they do at school, shoving them in our face with the reasoning ‘just because.’

If education is required to improve, then I will vote for this channel. The animation is superbly great which properly matches the movement of the eye, a great way to learn even with beginners and non-mathematicians. Moreover, to create videos like this, it takes a trench-level of understanding of the topic. What this channel is teaching probably isn't being taught in some schools and universities. It dives into the most fundamental concepts/roots and answers the derivation of formulas we learned in schools. You cant call math a beautiful subject instantly, but in this way, you can see that it is indeed extremely beautiful and interesting. Kudos to this channel and I am thankful that I am born in this era of technology.

the animation and teaching style is truly brilliant!!

This is an amazing illustration. Great job. I m so impressed by your clear explanation of such a difficult topic, that I have subscribed to your channel right away.

This is the best mathematics channel on RU-vid. There is literally no competition. I want these videos played at my funeral.

I know that it's been said before, but Grant, this is a stunning presentation. My heartfelt appreciation for your work can be assumed.

I love the illustrations and the animations and everything about it, I actually just subscribed too. Overall, I think you did a great job; additionally, it was better than what I could do. QUESTION: But, how you were able to get the area of the north and south "sections" of the slice of the sphere when they would be more triangular and not rectangular?

Geography teacher : doing 4 questions in 4 minutes is the same as doing 1 question in one minute Calculus : *am i a joke to you?*

This is exactly what I needed. Im going back over precalculus now that Ive finished trudging my way through calculus one (barely got a b+; havent been to school in 15 years and never really had a proper education ). I was going over the proof for the area of a circle and of cylinders spheres etc by extension and it finally clicked yesterday afternoon. This video has helped parse out all the details and cement it in. Perfect job.

your concepts are absolutely mind boggling. I will be subscribing to your channel now. Thanks for sharing it..

This is just what I wanted to know over 60 years ago, thanks!

If only the pen paper drawings were just as easy to play around with~

These videos are so beautiful, I've only just started pre-calculus in high school this year but I love learning more in my free time and it's so cool to see this video touching on some ideas I actually know some stuff about- and also it's presented in such an elegant, intuitive way that's so much easier to understand than anything my complete mess of a pre-calculus teacher could come up with.

What a great video! I used up have such a love of math but my ability to/memory of how to process it has definitely gotten worse and worse since i left school It was really cool to get such a thorough explanation of these concepts!

You know a video is good when it makes you do maths voluntarily lol, great job!

I'm now in my 40s and I find math more interesting than how I felt during my high school learning.

17 minutes of heaven.

I have never seen this explained in such a lucid and simple manner. Hats off to you sir.

I already understand the answer a fourth into the video, but wow i appreciate the amount of effort put into the animation and explanation put in this video. Truly one of a kind gem!

I'm torn. Is the math more beautiful or is it the animations?

I love this channel, I’m hoping watching enough of your videos will help motivate me in school. You actually explain these concepts from multiple perspectives ti actually grasp them

Thank you dearly! I'm not a student, I accidentally got into engineering. I just needed to replace my speakers power supply/cord, which a new cord would be expensive in relevance to the cost of the speaker. I decided to look up a little bit about electrical, math, and some science to repair it. 6 months later and some help from amazing sources like you I was able to work with an Arduino UNO! Math is a rabbit hole and so is all the subjects its beautifuly displayed in. It helps us to understand this abstract world. I see that school and I were not very capable. In fact I did not understand any of it. Failed alot but I have my GED. There is a lot of ways of learning and I found that I do not "space out". Im a visual person and that works for me when i let myself apply that type of learning and problem solving style. I find engineering is pretty visual and I don't think I'll ever be able to stop asking questions upon many subjects. These videos are very informational and insightful... especially into various ways of looking at something. Thank you! -cyborg with brown eyes 👀

your way of teaching is amazing. Keep it up...

This youtube comment section is gold. Thanks 3b1b for making such a wonderful community.

I've never seen such beautiful animations in my life!

I noticed that the 4 circles in the thumbnail is 3 blue ones, and 1 brown one, placed in spots to kinda resemble 3blue1brown

My God the level and details of animation!!! Thank you a lot...

I like how your explanation is mainly geared towards students in “pre”- calculus.

3Blue1Brown: "that's negligeable" Me: *dies*

I've always thought geometry is the best way to introduce many mathematical concepts. And why haven't I watched 3blue1brown before? This is very much my kind of explanation. However, as tired as I am, I might have to skip getting the paper out. I'll just have to watch this one again some time. :) Edit: Optical illusion at 12:08 -- when separated, the rings appear to shrink latteraly.

These videos and animations are amazing!!!

Good lord how do you do these animations? You're incredible.

"I mean viscerally feeling a connection between this surface area, and these four circles." And that's why this channel is so popular. If math was taught this way going all the way back to introduction of the base 10 number system, we'd probably be colonizing Mars already.

u r are such a genius ... i have never seen someone teaching like this , made mathematics much more fun.... this is fabulous i wish you to plz continue and try to make detailed explanation of some tough topics like schrodinger equation uncertainity principle,

My first video watched of your channel. . Liked. Subscribed. Awesome pedaogy. Very intuitive Animation. Thanks a million from India.

This has been literally the exact question I had in my brain since I learned this topic, 7 years now, and finally YT algorithm has found my inner self!!

these animations are astonishing. congrats.

amazing!! this is the best animation, thanks for your excellent job

You Are an extrodinary animator! Lessons from You being highly valuable. Learning math and geometry from You is a pleasure. Animation im sure would be too.

The video, the explanation, the simplification and the object of the work -- all are in point. This is how we need to learn maths. Mr 3B1Br, I'm really honored to watch your videos and the way this inspires me is inexpressible. Thank you so much!

The animations are absolutely brilliant.

that was brilliant man, well done, you made it really intuitive for everyone, keep up the great work

Man I've wondered this everytime I have seen this formula. Thanks so much 🙏 🙏 🙏

I've seen this video many times. Yet each time, its as exhilarating as the last one.

10pm: 1 last video and I will go to sleep! 3am:

I have to retake calculus this semester, and I have to say the part about seeing why limits or why integration really hit home and made it much less abstract thank you very much. I will have to watch more of these for sure.

Excellent presentation of the topics...very easy to understand..

Yayy i figured out the proof (challenge mode) ^_^ and what a great video Grant. Display of mathematical elegance. Love your work

I always used to love maths in school. And now that I saw mathematics in such beauty as you present it, I really start to miss it

Bro RU-vid. its 3 in the morning. Im not ready for math

I have a different take that I think is relatively simple. After answering the questions 1, 2 and 3: Question 1: C=2πRsin(θ) C×Rdθ=2πR²sin(θ)dθ Question 2: R(e):exterior radius R(i):interior radius cos(θ)=(R(e)-R(i))/Rdθ =>R(e)-R(i)=Rcos(θ)dθ C×(R(e)-R(i))=2πRsin(θ)×Rcos(θ)dθ =2πR²sin(θ)cos(θ)dθ Question 3: 2πR²sin(θ)cos(θ)dθ=πR²sin(2θ)dθ since sin(2θ)=2sin(θ)cos(θ) Therefore, the area of the shadow of a ring is half the thickness of the ring at twice the angle of the ring that casts the shadow. Questions 4 and 5: I decided to combine these 2. First of all, question 4 is partly answered by the conclusion of question 3. We see a shadow and when comparing it to a ring in terms of area we must skip a ring and look at the 2nd one as our answer. However, regarding the initial goal of proving the the area of a sphere is 4πR², we can take those rings and put them up against each other. Now, the shadow that's being cast will be a circle and as we proved earlier the area of a shadow is half the area of the ring at twice the angle of the ring that casts the shadow. However, since now the rings, since we added them all together, have become a hemisphere we can make the conclusion that the circular shadow that is cast by the collection of rings, now the hemisphere, is half the area of the hemisphere formed by the collection of rings. In addition, we know that 2 hemispheres make up a sphere, therefore the area of a circle, since the hemisphere casts a shadow of a circle, is a quarter, 1/4, of the area of the whole sphere: Sphere: S Area of sphere: AS Hemisphere: HS Area of hemisphere: AHS Shadow: Sh Area of shadow: ASh ASh=1/2×AHS 2×HS=S=>HS=1/2×S =>AHS=1/2×AS Therefore, ASh=1/2×AHS=1/4×AS And since a shadow is 2-dimensional and it's a circle since it was cast by a hemisphere, THE AREA OF A CIRCLE IS A QUARTER OF THE AREA OF A SPHERE.

jesus your videos are such a clean contrast to the sometimes messy world out there. and shows that even in the chaos there is beauty. thanks

great explanation with graphics! the best use of technology in teaching.

This is my favorite channel now

You must be a superstar just to pull out the animation in this video

Great video as always, just finished cal 2 with an easy A thanks to some help from your vids on Taylor series!

Could someone check if what I have is right? So the area of the rings for any angle θ is 2π(r^2)sinθdθ And the area of their shadows is 2π(r^2)sinθcosθdθ = (1/2)π(r^2)sin2θd2θ So for the shadow of a ring at an angle θ, there is a corresponding ring at angle 2θ having a quarter of the area. θ should range from 0 to 90, and not all the way to 180, because otherwise the shadow will be cast twice. Also the new sphere made from the corresponding rings has courser iterations. Although, now I think that (1/2)π(r^2)sin2θd2θ could have been expressed as π(r^2)sin2θdθ with alternate rings which I guess is what the video was going for.

Very good extraordinary explanations, animations are super. Thanks a lot.

Amazing way to communicate math ideas. I hope someday all school education will be done this way. It takes a lot to produce the content, but it is 100x more effectiveness and can make so many more students love math.