Understanding the Volume of a Sphere Formula [Using High School Geometry]

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Deriving the volume of a sphere formula. (4/3)πr(cubed) gives you the volume of a sphere, but where does the formula come from? Here is a simple explanation using geometry and algebra.
The volume of a sphere is the measurement of the space it can occupy. A sphere is a three-dimensional shape that has no edges or vertices. In this video, we will learn to find the volume of a sphere, deduce the formula of volume of a sphere.

[@mathematicsonline]

4πr^2 video explanation kzbin.info/www/bejne/bHbdgniad72SaZI At first the sphere isn't rounded because it is made of a few pyramids. The trick is to increase the number of pyramids while maintaining the same radius therefore the bases of the pyramids become smaller. By increasing the number of pyramids and the bases are getting smaller, so the collective shape of the pyramids gets rounder and it resembles more the shape of a sphere. If we have an infinite amount of pyramids then the volume of the pyramids eventually becomes equal to the volume of a perfect sphere.

[@aguyontheinternet8436]

Thanks for the surface explanation

[@ProofDetectives]

Asslam o alikum. Thank you for the video. Have a good day.

[@abelpalmer552]

I love how you concisely and accurately explained the concept of an integral without ever mentioning calculus.

[@obilisk1]

It's a really nice application of Archimedes' method for finding the area of a circle

[@emmawatson9180]

A R T

[@AlexandrBorschchev]

Ok nerd

[@backyard282]

You seriously in a nutshell said: "a quick explanation why the surface area of a sphere is 4pir^2 is because of the amazing fact that surface area of a sphere is actually 4pir^2"

[@mariopires8560]

that explanation requires another video, right?

[@AaronHollander314]

kzbin.info/www/bejne/fX_Gd518otZ4mZo

[@nyomanaryasejati2550]

Lol v: Here the explanation about surface area of sphere kzbin.info/www/bejne/bHbdgniad72SaZI

[@CASH-TO-THE-MERE101]

VidyVid 👀

[@swalbi1579]

Exactly my thought. You don't explain one thing by throwing another question into the room.

[@sumyiuli7803]

1:20 This is the most beautiful thing I've even seen in a math channel.

[@howtungchong174]

agreed

[@goncalofreitas2094]

Me too

[@ksrinivas1729]

me too

[@sujitujitbista2379]

SUMYIU li

[@AaronHollander314]

Check out 3Blue1Brown

[@melonenlord2723]

Volume and Surface of sphere with Integrals: Koords of a sphere: x=R*sin(a)*cos(b) a=[0,2*Pi] b=[-Pi/2,Pi/2] y=R*cos(a)*cos(b) z=R*sin(b) derivatives of a: x=R*cos(a)*cos(b) y=-R*sin(a)*cos(b) z=0 derivatives of b: x=-R*sin(a)*sin(b) y=-R*cos(a)*sin(b) z=R*cos(b) crossprodukt from derivatives of a and derivatives of b: x=-R²*sin(a)*cos²(b) y=-R²*cos(a)*cos²(b) z=-R²*cos²(a)*cos(b)*sin(b)-R²*sin²(a)*cos(b)*sin(b) =-R²*cos(b)*sin(b)*(sin²(a)+cos²(a)) =-R²*cos(b)*sin(b) norm n of this vector: n²=x²+y²+z² =R^4*sin²(a)*cos4(b)+R^4*cos²(a)*cos4(b)+R^4*cos²(b)*sin²(b) =R^4*cos²(b)*(sin²(a)*cos²(b)+cos²(a)*cos²(b)+sin²(b)) =R^4*cos²(b)*((sin²(a)+cos²(a))*cos²(b)+sin²(b)) =R^4*cos²(b)*(cos²(b)+sin²(b)) =R^4*cos²(b) n=R²*cos(b) calculation of surface area A and volume V of the sphere with integral: A=int(int(n,a=0..2*Pi),b=-Pi/2..Pi/2) =int(R²*cos(b)*2*Pi,b=-Pi/2..Pi/2) =R²*sin(Pi/2)*2*Pi-R²*sin(-Pi/2)*2*Pi =2*Pi*R²*(sin(Pi/2)-sin(-Pi/2)) =2*Pi*R²*(1-(-1)) A=4*Pi*R² V=int(A,R=0..r) =4/3*Pi*r^3-4/3*Pi*0^3 V=4/3*Pi*r^3 Yea, i'am bored and have no gf

[@SoYuan]

wtf.... wow

[@MrHenryG123]

you could also just say volume is the integral of the surface area since he jut blatantly uses the surface area formula here

[@christinadc]

Philipp Petsch Have u copied or written it yourself??? OMG !!! I was just shocked. I think you could be the boss of mathematics hmmm... 👍👍👍👍😊😊😊😊😊😊 But I haven't understood even a bit(Lol)😕😕😕😕😕😕😕😕

[@melonenlord2723]

Christina DC thank you for the compliment. :) Didn't copy it, but when you know how integrals work, then it's not that hard. Had to learn this staff last year and so i know it. But please don't be sad, at first, for me it was hard to unterstand too. It takes some time and explanation. I only gave formula here. I can try my best to explain some more of what i did here if you want, but I can't guarantee that it helps, because for me it's not easy to explain something in english. It's not my main language and google translate sucks sometimes. :D

[@coolguy4989]

Philipp Petsch did you seriously just calculated how to calculate the area of a sphere with trigonometry in a KZbin comment section? Man you must be really bored

[@calvinvondracek1418]

Day XX of quarantine, learning how to find the volume of a sphere

[@יעקבלוי-ו9ל]

You will be einstein by the end of it

[@blackcat5771]

@@יעקבלוי-ו9ל lol

[@jamesdesantis9420]

We all gotta keep busy!

[@rickdearmont9950]

Exercise the mind!

[@puretwr899]

Me too !

[@alanzzstyle6805]

I don't know a lot of english myself and i understood your lesson 1000000000000x times better than any other video in my language, 1000^999999 times thanks!

[@djzanis5220]

0:26 this version of interstellar brought to you by mathematicsonline

[@avidrationalist7281]

This is what real mathematics is. We were so unliky that our teachers never demonstrated these concepts while were were learning at college. I love maths from my childhood, but our teachers were unable to show us the beauty of mathematics, and I lost the chance to build a career based on engineering subjects, and ended up becoming management graduate. However, there is a saying that you are never too late to learn something...so now I am learning these amazing techniques and teaching students these beautiful techniques of mathematics...

[@hero9402]

I was really ashamed inside that I didn't know why the formula of sphere's volume is that. I thought they taught but I forgot how they derived it but it turns out, they did not. They just made us memories the formula. This is so dope thank you.

[@mathematicsonline]

If we have an infinite amount of pyramids then the volume of the pyramids eventually becomes equal to the volume of a perfect sphere. And I do have a video response explaining the surface area of a sphere, thanks for watching.

[@anilkumarsharma1205]

where are tangent theta ,please tell me about it🔬🔬🔬🔬🔬🔬🔬🔬🙏

[@lelouchlemprouge6380]

why cant we integrate

[@CalculusPhysics]

Anime Movies and series This is explaining what the integration is doing. most people have no idea what integration is, and even if you do, just showing an integral isn't helpful or intuitive

[@emmanuelkuzniak4534]

Great video. But even if we have a lot of "pyramids", they are never real pyramids because there is always a very small round part at the base. Where is the sum of these tiny round parts in the formula ?

[@camillefailes54]

@@emmanuelkuzniak4534 I think that's why calculus eventually gets involved, to add up all those little pieces (pretty much integration, but not precisely)

[@abhijeetkishore1412]

I'm mesmerized. I was looking for this kind of explanation from my childhood. Thank you so much.

[@jessedeleon7392]

I love your animation. Surely, students will understand how the formula is obtained. I wish you would make similar videos for other formulas.

[@lexinaut]

Very nicely presented! Visuals are quite awesome, illustrative, and creative. Spheres are quite fascinating, even eerie, which is why they are sometimes called SphEeries. Similarly, circles are often called Quirkles, because they are quirky! Of course, Calculus provides more rigor in solving this vital volume problem, but the visuals here have plenty of helpful VIGOR to help people get to more RIGOR! Thanks for all your dedication to edication (oops, education)!

[@raffichilingaryan389]

Lol "and the amazing fact is, the surface area of a sphere is equal to four of these circles!"

[@X225-c3d]

Yeah but how ?

[@ketanpandeySynerge18]

A VERY GOOD EXPLANATION FOR THE DERIVATION OF THE FORMULA OF VOLUME OF A SPHERE.IT IS THE BEST METHOD A BEGINNER CAN UNDERSTAND THE DERIVATION OF VOLUME OF A SPHERE FORMULA. EXTREMELY USEFUL FOR STUDENTS.

[@officialkian7317]

Im 14 and my maths teacher wanted usto make a presentation about this and this video really helped me understand the steps. thanks man x

[@TahirKhan-jq7qv]

Same but I'm 10

[@tristanhidalgo8463]

Same but I'm 3

[@Saltofreak123]

I'm a fetus

[@KamuStoku]

I am a sperm

[@ForfunckleStudios]

I am the parents of myself

[@miyari26]

2年ぶりに見つけた、、！ This video is easy to understand!! Thanks☺️

[@nberz692]

The best explanation can be shown with integrals

[@chrislor2882]

This one its better, its more intuitive

[@Leonardosanchezandre]

we are stupid

[@vinodkumar-wm3oq]

Great visual proof sir! I like the ways you derive the formulas by simple methods that anyone can understand. Good job

[@zord3970]

Wow, this is as great explanation. Thanks! I like knowing where the formula comes instead of just blindly accepting it.

[@youfrancis]

Just wanted to say: this was an EXCELLENT explanation, and superbly animated. You've done a really great thing here--thank you for making this!

[@xxphoenixx8398]

Came across this video while researching for an assignment, and wow, your channel's very interesting!It seems like you explain the steps behind the formulas,and I really need that to solve math problems.

[@rickybobby5584]

amazing work man but you didnt explain why the surface area of the sphere is 4 of the circle area. but thank you any way for your efforts.

[@TonyStark799]

+ricky bobby That's not the topic of the video.

[@Pankaw]

There's another video where he explains that

[@realityversusfiction9960]

Elementary Foundation Stone Of Differential Geometry Given a 120 centimeter Diameter multiplied by three, the length of the circle will be 360 centimeters. Given 360 degrees to the circles length, each degree will be one centimeter in length. And not one or any number of mathematicians, geometer's or geniuses can alter or change the fact that; A circles length is exactly three times that of its diameter length. Area of a 120 centimeter diameter Circle 3 X r2 (Three times the radius squared) 120 cm x 120 cm; Square = 14. 400 sq cm 1 right angle = 120 cm x 120 cm ÷ 2 = 60 cm radius 60 cm x 60 cm = 3,600 sq cm 3,600 sq cm x 3 = 10, 800 sq cm to the area of the Circle Which is three quarters of the area to the 14, 400 sq cm square. In sum A Circles Area is always three quarters of the area gained by squaring its diameter length. A circles length is always three quarters of the perimeter length gained by squaring its diameter length. Twelve Steps From The Cube, To The Sphere (Volume & Surface Area) Calculating the surface area and volume of a 6 centimeter diameter sphere, obtained from a 6 centimeter cube. 1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimeter’. 2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm. 3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm. 4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity of the cube = 54 cubic cm. 5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm. 6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm. 7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm. 8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE 9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm. 10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere. 11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm. 12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere Confirmation by Weight Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grams, prior to it being turned on a wood lathe into the shape of a sphere The Cylinder of the Cube would weigh 120 grams The waste wood shavings would weigh 40 grams Given that the Cylinder weighed 120 grams The waste wood shavings would weigh 30 grams. Note: And ironically you can also obtain this same result by volume, using Archimedes Principle. www.fromthecircletothesphere.net

[@k0pstl939]

he did that in annotated video

[@F34RFACTORR]

Yes he did... Watch it again man, and all the way through.

[@JuanVanSteyvoort]

Brilliant !.... Why haven't I got a teacher like you, when I was at school ?..... You are fantastic.... Thank you so much for that magic video... :-) From Brussels, with Love....

[@theintelligentcomparison]

This is really good . Enjoyed your explanstion

[@grumpyparsnip]

Bravo! Beautiful high school level explanation of how to derive the spherical volume formula from the surface area formula.

[@mathematicsonline]

Glad you liked it

[@MrBrix3]

Waiting for explanations like these for a long long time....................Thanks Best videos ever!! Wish this concept could be applied to other areas of study!! Pure Genius Thanks so much.

[@geraldillo]

I think that this is just a (nice) algebraic trick because you can't divide a sphere into equal sized square-based pyramids.

[@AbdulhameedAlMahfadi]

It's very helpful and nice. May I ask you, which Programm do you use to do the graphics?

[@pabloduartedossantos6705]

Thank you my good sir, you explained it so perfectly, thank you very much

[@somyaagrawal4120]

You are the best maths explaining teacher. 🎉🎉 very nice video. Thank you so much😊😊😇😇

[@drfleau]

I totally agree with your method, which is exactly what I was saying, the calculus (integration) method is the most complete and thorough. The limit of the length of one side of the base is 0, eliminating the curved surface effect. We agree ! :-)

[@AlbornozVEVO]

I loved the explanation for the surface area formula. First time I see a proof by amazement.

[@j5v133]

for a class 9th student, ur videos for surafce areas and volumes are perfect.....amazing videos

[@walding7258]

Thank you, the lesson gives me understanding of the subject.

[@leng3195]

T6hanks this was helpful although I had to watch the video twice to understand to anyone who is still confused after watching it the first time hit the replay button and go to 3:00 exactly and he will explain everything that may have confused you again in simpler terms

[@mrudulasrinivas5738]

It was amazing and very easy to understand.............Truthfully speaking it is very very interesting . hats off.................

[@ian5576]

I agree with your assertion about surface area proof is 'lacking' however this is the best non calculus solution I have seen for deriving a spheres volume. "Moreover, the bases of the pyramids are not flat, but have a curved surface", That doesn't matter as the sum of all bases are substituted with 4Pir^2 which is curved, he has essentially done a limit by setting their sum equal to this. Calculating these things with calculus is commonplace even though the formula has been around much longer

[@arroz2092]

Love this channel. I learn each day something new.

[@SabrinaXe]

Your visualisation is beautiful, keep it up!

[@Roberto-REME]

Outstanding explanation and your graphics are very helpful. Well done!

[@alxjones]

@vampiracy Just as a note: sqrt([F_φ]^2+[F_θ]^2+1)=1. This is because we're talking about a sphere of radius R. This is represented in spherical coordinates as ρ=F(φ,θ)=R. Since R is a constant, taking it's derivative with respect to any variable will yield 0, so [F_φ] and [F_θ] are both 0. When we plug this into our surface area differential, we get sqrt(0^2+0^2+1) or sqrt(1), which equals 1.

[@fontanot]

Thank you fo explaining. You just earned another subscriber

[@yangxxuan0113]

This helped a lot! As a 6th grader I was able to understand it thank you!!!

[@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

[@Hereford1642]

TroyaE117 You are right but your method relies on differential calculus. The method and reasoning shown in this video is that which was used by the ancient greeks long before Newton and Liebnitz.

[@edwardsuleski]

So did Archimedes or Pythagoras invent this idea?

[@Hereford1642]

+Edward Suleski Wikipedia - history of calculus credits Eudoxus who predates Archimedes by 100 years. Its called the method of exhaustion. Everyone has heard of Archimedes and Pythagoras so they get all the credit while poor old Eudoxus is flagged as a spelling mistake. Not fair eh?

[@chrissekely]

+TroyaE117 AS jack hodges pointed out, your solution requires calculus. To most people, calculus is *not* simpler. Therefore, your solution, though correct is not simpler than the one presented in this video. No offense intended. Just pointing out what I would have thought to have already been obvious.

[@TroyaE117]

+Chris Sekely Chris, If you think the method in this video is simpler, I suggest you brush up on your calculus.You will find it very interesting, transparent, and very much faster.Best...Troya

[@DanielKatundu-y8t]

This channel is just absolutely beautiful.

[@etlewis69]

This is why I failed geometry in high school. I watched the video twice and it still makes no sense to me. I never had a problem with algebra, but geometry just makes my mind shut down.

[@shahbaazali1195]

Superb work guyzz, finally found the type of explanation that i dreamt of but never had experienced , thank you soo much for making it mathematicsonline

[@mattyblake9427]

Spectacular demonstration. Math is beautiful if you understand it. Using the pyramid amazed me ❤

[@taeseonmun7056]

How wonderful idea!! If we can think of this idea, we don't need to do any (inaccurate) experiment. Thank you for sharing.

[@tombuis4438]

Excellent! Visuals, explanation, pacing, structure of the presentation - I can think about spheres differently now thank you so much 🎉👍🍻

[@hemanshutube]

Marvelous, Never seen like that...... good work Thanks for spread this knowledge world wide through world wide web. Excellent from my side.

[@divyamagrawal2614]

An actually good video my ma'am made our class watch

[@janisjuplin7817]

Thank You for making math formulas vivid and clear

[@hassantj5775]

A very brief & to the point explanation. Keep it up!!!

[@michatroschka]

extremely well explained, and its even entertaining and calming

[@dragsterbixing2584]

I think you just loved the animation there is no sense in the proof though

[@droidinator]

More on the last few sentences. The volume of the cone is 1/3*pi*r^3. Caliveris states that shapes with the same cross section have the same volume. Therefore, with similar volume, it is possible to use the subtraction above. Otherwise, just integrate pi*(r^2-x^2)) from r to -r. This is the integral equation for volume (pi*r^2). This works since integrals are the some of all the cross-sectional disks and the equation of a circle is all the points create the radius.

[@dilipgupta474]

wow wow wow.....n times wow......my heartiest wow to your process of explaining as well as the animation 👍👍👍👍

[@alxjones]

@Kosekans Okay. We'll integrate unity over a region in 3D space, E. E is defined as a sphere of radius R centered about the origin. Integrating it will give us the mass of a sphere with radius R and uniform density of 1 throughout, also known as the volume of said sphere. We'll need to set up a triple integral in order to solve this problem. We'll be working with spherical coordinates for simplicity, since we have a spherical region.

[@samytanjaoui8178]

Thanks for this great work. With which programm did you make this animation?

[@MooMooMath]

Thanks for the video. Love the graphics. Well done

[@nicolita222222]

I love proofs.

[@dineshmutha4475]

surface area of sphere cut at h height from crown

[@liliacfury]

Same!

[@ambadasdoddi6968]

Very nice

[@CASH-TO-THE-MERE101]

Nicole Melendez Pena 👍

[@ripinpepperonies9754]

V lmao

[@chenadam6339]

Really appreciate your visualizing work, it makes me understand a lot better!

[@uttemrajigadwe7145]

Thanks for this very lucid example

[@zactaylor6676]

Just learned more in a few minutes than I did in all of high school math. If only teachers could have explained it like this and not to like 30 other kids at the same time

[@G4VRX]

Thankyou . Nicely animated. I liked the drole ... area is B1 plus B2 plus B3 ! Archimedes and projecting a sphere area onto a cylinder can prove the formula for surface area is 4 Pi r squared . You used square based pyramid bases which visually fit nicely, but curiously this process also works out the same formula using thousands of radius high CONES instead of thousands of radius high square based pyramids. Both base shapes and any errors are irrelevant because you suddenly swop the real result of the integration for the previously worked out area formula. NEAT .This is definitely early calculus, did the ancients do it this way ?

[@ickywitchy4667]

In conclusion...after watching this i forgot the perimeter of a triangle

[@HAQ222]

Hahah

[@ytuser4562]

omg sameeee

[@Sourav1973]

😂😂😂

[@movieentertainment2486]

Sir you are doing such an amazing work, I hope you can reach over infinite subscribers

[@megaliora]

thank u a ton ur video truly helps me with understanding maths and solveing problems & i really enjoy watching it becauce they are so interesting to watch and made me imagine all the details , I'm so thankful hope u all the best

[@PhilemonXIV]

Clear, easy to understand, excellent presentation...very cool :-) Thank you!!!

[@TheSparklingLavender]

Video was wonderful but u have to watch this more that 1 time to understand each part easily....but nice video..keep it up!

[@munilalnayak2600]

I love your videos Thanks for fabulous work We could ascertain which we can not imagine alone.

[@chrissekely]

+mathematicsonline I enjoy math. I have a math minor from college I've never used and I watch Numberphile often. I also really like your derivations. I am curious about one thing. Way back in high school (or maybe early college) I thought up a question I still haven't seen answered anywhere. If one were to cut a hole through a sphere and supposing the hole had a perfectly *square* (not round) cross section, how would one calculate the volume removed (or remaining)? Assume for simplicity that the central axis of the hole coincides with a diameter of the sphere. I'm sure this could be done with some relatively complicated calculus ( relative, at least, to the complexity of the derivations presented here). However, I'm hoping for a more intuitive approach, provided that such is even possible.

[@lucasm4299]

A very good question. Yes, of course it is possible to find the volume with some good advanced calculus. I do not know the answer at the moment but if I figure it out, I will let you know. Volume of Sphere: 4/3 pi r^3 Volume of Rectangular Prism: 2rs^2 4/3 pi r^3 - 2rs^2 < ACTUAL remaining volume I would need to find the volume of a rectangular prism whose ends are part of the sphere's surface area I do not know how to call that shape but then you would subtract that from 4/3 pi r^3. This might be a problem too hard for me. I don't know how functions behave in 3-D. Good luck.

[@niyatitejwani]

I understood it . Nice video. Thanks for explaning . I actually wanted to know that how the formula derived .

[@dylanmewett9164]

bro all other people use integrals and i see this my savior

[@mathematicsonline]

Lets assume the sphere isn't rounded because it is actually made of pyramids. The trick to make the pyramids equal to a sphere is to increase the number of pyramids while maintaining the same radius. What I mean by increasing the number of pyramids is by making the bases of the pyramids smaller. So imagine we are increasing the number of pyramids and the bases are getting smaller, as a result the collective shape of the pyramids gets rounder and it resembles more the shape of a sphere.

[@erahamzah6983]

omg, the dissection but so easy to understand jeezzzzz

[@MrUser6785]

Thanks so much for the video....this is the first explanation about this that I have understood.

[@sherparoyale]

The challenge of this proof is that is requires “believing” that surface area = 4 x pi x r^2. My daughter (in 8th grade) just shared this proof and I explained why the proof inherently had an underlying assumption that she couldn’t prove. Calculus and a volume integration is a far better way to prove this.

[@dineshrajoria8066]

This is quite an amazing explanation👍

[@tearchi]

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

[@mohamedalserali7843]

We can take volum of revolution of curve between (x,y)represent by equation of y^2=r^2-x^2 and then take Integration to give you half volume of the sphear

[@jimsaheb]

To respond to Daren Soobrayen: Good observation. This explanation might help: When I studied calculus in undergraduate school, they taught us to add together an infinite number of squares to determine the area of an irregularly shaped object. So imagine a flat surface such as a butterfly shape drawn on a piece of paper. You want to determine the area of that shape. If you draw an infinite number of squares and/or rectangles on that shape, then add together all of those areas (which are easily determined by using the formulas for a square and/or a rectangle), you can then compute the area of the butterfly shape. So looking at the sphere, imagine an infinite number of pyramids each with a base the size of a pin point; or imagine pyramids the size of a human hair. Then add all of the bases (rectangles) together, and you come up with the surface area of the sphere. Just make the pyramids' bases so small that they fill in every nook and cranny of the curved surface area of the sphere. I hope conceptualizing it like this helps out. If you make the bases (and the pyramids) small enough, they'll fill in even a curved surface.

[@awZomePig]

Same thoughts, (found a mathematical proof for the surface formula, tho - rly hard ^^ ) but i kinda worked out the "curved surface" thing: Lets just simplify it and say that a "sircle" is a figure with "infinite" sides. It's not a number. And the amount of pyramids are also infinite. But there is nothing like a perfect sircle, as when you zoom and zoom you'll find irregular patterns in the atoms, etc.. So the sircle isn't round, and the base of each pyramid represents each irregularity

[@alxjones]

@vampiracy Our limits of integration are φ=0 to φ=π, and θ=0 to θ=2π. Integrating sinφ with respect to φ, from φ=0 to φ=π, gives us 2. Integrating 2 with respect to θ, from θ=0 to θ=2π, gives us 4π. Multiplying this with the (R^2) we took out gives us a surface area of (4π)(R^2).

[@monicadupont9606]

Excellent! It helped me a lot. Thank you.

[@wv1seahawks]

You could also do it by cutting the sphere into infinitely many shells (hollow spheres.) Each of which has a surface area equal to 4(pi)(radius of the shell)^2. Then the volume of any shell would be equal to the surface area of it times dr (the infinitely small width of the shell.) then the total volume of the sphere would be the summation of the volumes of these shells. This would be equal to the integral of the surface area equation with respect to r from 0 to r. This integral is equal to (4/3)(pi)r^3, the volume of a sphere. I just appreciate the fact that volume of the sphere is equal to the integral of its surface area

[@bonbonpony]

+wv1seahawks Infinitely thin shells have no volume. You cannot get a volume with adding 0-volume objects, no matter how many you will add up (even infinitely many). This is a common misconception that haunts Mathematics from centuries: that volumes are made of surfaces, surfaces are made of lines, and lines are made of points. This is utter bullshit, because their dimensions don't match. Points don't have length. Lines don't have breadth. Surfaces don't have volume. (Remember Euclid?) True, lines _contain_ points, but it doesn't mean that their _length_ is made of the "infinitesimal lengths" of those points, because again: *points don't have length*. True, surfaces can _contain_ lines (and points as well!), but can we say that the _area_ of the surface is _made of_ those lines? (or, even worse, from these points?) :P And one cannot get something from just adding up nothings, even infinitely many. This is not where all those dimensions come from.

[@ripperfisher182250]

+Bon Bon no it is common to use shells to find volume. you are wrong here m8.

[@bonbonpony]

***** The mere fact that I disagree with the majority, doesn't yet make me wrong. And the fact that something is commonly used, doesn't mean yet that it is correct. Even the fact that it seems to "work", doesn't mean yet that it really works this way, because it can also work "by accident", from different reasons. (From false assumptions you can derive everything, be it true or false.) As I said, infinitely thin shells are surfaces, they don't have volume. So no matter how many of them you "add up", the sum will still be 0. Notice that when you're calculating integrals, you're not really doing the summation directly. Usually, you're not even doing the limit directly. You just apply known formulas. These formulas were known before calculus has been "invented" by Newton, Leibniz & co., because they've been known from antiquity, and they've been derived without any notions of infinitesimals, limits, 0-thickness shells etc. The Greeks have got them from the Egyptians and from the Arabs, the Arabs got them from the Hindus, etc. Unfortunately, just the formulas, without their derivations. So Newton & co. needed to "reinvent" these derivations to be able to pretend that they invented calculus all by themselves :P They had to "made up a theory for the formulas", and they did it wrong. But the deception worked so well, that it works to this day, since people are being taught these techniques in schools and being taught to not question it (since questioning usually lead nowhere anyway, because tachers don't know the answers too :P ).

[@fahimcan12345]

As far as I understand it, it's not that you're adding 'volumes' with "0-volume objects" (even though such a volume is just 0). But the idea is explained either with infinitesimals or limits with which you are looking at the volumes as the depth of the strips are approaching 0 or are extremely close to 0 (but not exactly 0). An infinitesimal is something that is extremely extremely close to 0 but is not exactly 0 and so anything with a depth that is infinitesimally small still has a depth (so to speak) but it's just that it's very close to 0. For example, when differentiating the principle we're always taught is using limits. And so we look at what happens to the gradient of a function at a particular point. Therefore, we take two points on a tangent line at that point. The idea of the limit is just looking at what happens when the vertical distance and horizontal distance approach 0. This doesn't mean you're doing the calculation 0/0 but rather doing calculations that get more and more accurate the smaller the distances get. And anyway, it seems more to me that we just use this idea to derive other formulae and not actually on it's own to calculate stuff.

[@ripperfisher182250]

The fact is that what he just said is another way to explain summation of infinitely small objects to estimate the volume of one object by cross sections or cylindrical shells. ECT. Many methods however all he did was rephrase the process if you don't understand it's because you don't want to be wrong and you are just being rebellious. Zz. To argue this point is just wrong because by definition the process is logical.

[@Boomshicleafaunda]

These in depth proofs are awesome.

[@mathsease1503]

Enjoyed the explanation and graphics

[@RaushanKumar-uw7qb]

This video is really amazing.

[@SK_MDHS]

Thank you so much! You explained everything in detail

[@ssuresh3012]

Wonderfully explained and very nice presentation.

[@schemjong1299]

Brilliant visuals! Thank you! I am sharing this with my students. :)

[@fwcolb]

Volume sphere = V = Pi * 4/3 * r^3; Surface of sphere S = Pi * dV/dr = Pi * 4 * R^2 Circumference of disc through center C = Pi * dS/dr = Pi * 2 * r Problem is that when you differentiate, the constant Pi is lost. And when you perform the operations, you have to add Pi. Notice the topology: volume, surface and circumference. Surface area of sphere-> circumference of the disc is intuitive. But volume of sphere -> surface of sphere is less intuitive, at least a priori Strikes me as a bizarre approach to the question but it just popped into my mind.

[@meow3343]

Understanding how this thing works is way better than memorizing all the formulas in the Geometry

[@امانينا2022]

That's the most beautiful thing I ever seen❤❤❤

[@renciebongalon228]

Great explanation........the learner would really appreciate the video...more videos please...thanks and God Bless

[@Erebus2075]

this was soooooooo goood!!! so easy to follow and understand like this TY

[@renecabuhan1675]

Amazing visualization sir.