Understanding the Surface Area of a Sphere Formula 

***** Just intuitively, humans have been doing that in math long before calculus. One thing that is cool is that mathematicians were close to inventing calculus back in the ancient historical times. I forgot the details but I think I read it in a Lancelot Hogben book.

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

Kemerover He didn't use integrals did he? I only watched the surface area of a sphere part. Oh when you said cone I started to think about volumes sorry. Anyway not all formulas require calculus to derive them. Some formulas were derived about 1500 years before calculus. Some of the concepts that are used in calculus were developed about 2,000 years ago.

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

Ivan Ereiz Yes he just used one of the concepts used in calculus but he didn't use the 'language' of calculus so this is a nice piece of work that everyone who has high school algebra can understand.

how am i supposed to understand this stuff?

@@flower_girl4983 learn the basics first for example the properties of triangle and other basic shapes, then go for average (this video) and finally the difficult ones. That's how you can master the art of learning mathematics

@@sam-ui5lc ok sam thnks a lot

@@sam-ui5lc and limits too

Thnks

Check out the 3b1b episode too

I relate 100% to your comment!

The better way to spend your time is to learn calculus.

@@zachansen8293 yeah that would have been really easy during o levels man thanks bro should have just done that on top of studying only 2 months

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

Hermes Mercury simple mind?

Simpler than integral calculus.

Well it wasn't THAT hard to understand :P

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

I totally agree.

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

dekippiesip its*

dekippiesip approximation*

dekippiesip gets*

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

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

Thanks

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

@@stephyelle1 his "proof" was more of an experiment test by comparing the volume of a cylinder with the volume of a sphere plus a bicone. There's a numberphile video about this

Thnkx

Glad to hear you enjoyed it!

@@mathematicsonline Thanks for sharing but I just don't see how or why anyone would cone up with this at all? Especially since it's so convoluted and unintuitive. My idea for a proof is 4 pi r squared is 4 times the area of a circle so you can think of a sphere as having four "faces" like a box has four faces. So you can thinkof a sphere as made up of four 2d circles projected into 3d space and hence the area is 4 times the area of a circle. This seems to me like a valid alternative proof?

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

Very nice bro

Hey...I'm losing you. "To form a sphere, you must rotate a circle around its diameter." Okay, that makes sense. "The surface area of the sphere is equal to the circumference of the shadow times the distance it was rotated." Again, that makes sense - and the circumference of the shadow would be equal to the circumference of the circle. "So we plug in: “SA=2πr*o” in which o is the distance the circle was rotated around." Got you. "Now, if we look AGAIN, we can see that the distance it was rotated around was actually equal to the diameter." Wait a sec...why is the distance it was rotated around equal to the diameter? If I have a circle, and I rotate it by 180 degrees with a diameter of that circle as its axis, and let the points on the edge of the circle trace out a surface, points on the part furthest will have moved pi r, and points closer will have moved less...how did you get that the distance rotated around was equal to the diameter? Different points on different parts of the circumference of the circle rotate by different amounts.

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

@@gligoradrian784 What distance is constant?

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

Thnks

Thnks

That's an even simpler mnemonic tool.

thanks Ryan.

Ryan Bell thnx

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

A= 4pi*r^2.... r= d/2....... so 4*pi*(d/2)^2 => 4*pi*(d^2/ 4).......4's cancel and all you have left is pi*d^2. Hope this helps.

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

Nyx Avatar what is calculas

You're right, however, as someone who's curious but not up to calculus yet, I really appreciated this proof. It was simple and only required a decent understanding of geometry and manipulating equations, making it more accessible to a far wider audience.

But how do you derive the volume? Btw what you stated isn't always true, for example in a cube the rate of change of the volume is only half of the surface area, cause increasing the side only affects one direction, which would be analougous to the derivative of the sphere volume with respect to d

I always get slightly confused when I think of it this stuff using derivatives. Like if you differentiate a circles area (pi r^2) then you get 2 Pi r - the circumference. Differentiate that and u get 2 Pi, the amount of radians in a circle. But what happens when you differentiate that? What’s that? And when you differentiate a spheres volume, you get the surface area, differentiate that and u get 8 Pi r - the circumference of a sphere??? It just leaves to many loose ends...

Indeed. The formulas are probably true when there are more than 2 frustrums, but that needs to be demonstrated, which it wasn't.

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)

Hi like im dad

Damn.

What?

I lost you at 'Holy'

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

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

Archimedes didn't have integral calculus.

Appreciate it!

They work for any polygon with an even number of sides

What about surface Integrals?

+VivzStudioSs "Double integral"

VivzStudioSs Surface integral turns out to be a double integral

It takes an understanding that the differential surface area of a simply bounded function in three space is linked to the sum of the magnitude of the cross product along two bounded axis. The double integral with the differentials alone is the area, and the function defines the height at each mid point. You need to understand partial derivatives to understand what I'm talking about. Deriving the surface area of a sphere is straight forward, by what if you have a simply bounded elliptical hyperboloid with specified boundary parameters?

@@guitarttimman Take two coordinates (rcosx , rsinx) , people don't realize that if sinx is increased by r times then that new point , rsinx , is increased by r in the x direction. So before integrating all points on the circumference , rsinx must be multiplied by r again giving us r^2sinx. Then it's easy to integrate to get surface area take the definite integral of 2pir^2sinx over o-pi.

Yes that would better generalization. It is curious that the derivation fails if it is an odd-gon. It has to be an even sided polygon for it to work.

Thank you for your comment!

Thnks

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

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

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.

William McCormick So you're trying to say that everything we know about the number pi is wrong?

Not everything however the super pi is going in the wrong direction. Sincerely, William McCormick

I think that he's trying to say that most real surfaces are imperfect enough that measuring them with incredible precision doesn't get you closer to their real value. Of course, this doesn't mean that knowing the precise value of pi is useless or that there aren't objects in our universe that require precision greater than the estimate of 22/7. If he's saying anything else, I think he's a crackpot.

+Abel Feltes I might fall under the latter, in your opinion haha. Archimedes was well ahead of his time, perhaps even our time. According to the old pre-World War Two teachings in America, Archimedes as a very large fellow nearly seven feet tall. Archimedes spent a good portion of his life working on a marble wheel. Each time he increased the accuracy of his marble wheel and the marble slab he rolled it on, it increased the length it traveled in one revolution. I had a similar result. I had done the test to finally see for myself what the truth was. I had machined a wheel out of a slab of 70-75 T-8 aluminum. And when I was done I rolled the wheel on a slab of the same material. To my disappointment it rolled a distance that matched the diameter to a ratio of about 3.14159 which meant i was wrong. I am a very good sport so I cleaned the wheel and the slab with xylene, and rolled it to get the most accurate measurement I could, so I could report it to those that were interested. And that is when I was rather astonished. The same wheel now on a nearly sterilized surface, rolled a distance to diameter of about 3.14308 I suddenly realized what Archimedes had been discovering. Most people believe what they are taught, no matter what it is. Having grown up in an Aero Space family, I can assure you much of what is taught today as truth is just fantasy. In my opinion this modern value of pi as well. Most people believe that 22/7 is much too large a value for pi. However from other master machinists, as well as my own machining of an exact wheel and base, I believe pi is slightly larger than 22/7. It is easier to believe if you see how metal expands more than it contracts when rolled or bent. I have manufactured a lot of pipe from flat stock. And I have done a lot of work with rounds and bending things into round shapes. So I know the measurements before I start rolling or bending, and the measurements of the finished product. I have conspired with other programers to create computer programs to calculate formulas for bending material, using these real changes during bending. So I am not a newbie to this subject or just a troll getting attention. I really love the real universe and how it actually works. After all, these are all the laws of God that you and I cannot break, his laws of science, the laws of the universe. We can misreport them however they will be there for all eternity. Of late I have found that many are stuck on one view they learned about circle math and pi. When in fact there are better, easier and more accurate ways to do some of these calculations. The actual universe works because of ratios between things large and small. For instance the reason a cell cannot become larger is because its rather spherical shape prohibits a larger cell from having enough cell membrane to pass waste out of the cell, and food into the cell to feed a large volume with a rather small surface area ratio. The smaller cell has a much larger surface area to volume ratio than the larger cell. The formula for surface area of a sphere is pi times diameter squared. The formula for volume of a sphere is pi times diameter cubed divided by six. Today they teach the formula for area of a sphere as 4 times pi times radius squared. It is basically the same thing however it seems to introduce possible error. Today they teach the formula for the volume of a sphere as 4/3 pi times radius cubed. And again this formula appears to complicate the simplicity. It also appears to introduce error and slight problems with base ten. If you look at a cube and a sphere that fits exactly inside it. You realize that to get the area of the cube we must multiply the length and height of the cube and then multiply by six to get the area. For the sphere inside that cube, we must multiply the same length and width of the cube and then multiply that by pi, whatever you believe pi to be. So the ratio of area of that sphere, to the area of the cube is 52.35987755982988 percent. Using the currently taught value of pi. Now if you look at that same cube with a sphere that fits exactly inside of it. You can see that the volume of the cube is length times width times height. And the formula for the volume of the sphere is the same length times width times height, divided by six, times pi. And again we see that the ratio of the spheres volume to the cubes volume is 52.35987755982988 percent of the cubes volume. Basically the ratio of pi/6 is the ratio of the sphere to the cube. The ratio of the surface area of a sphere and the volume of a sphere is also the secret to curing most cancers. This was actually taught in some schools before 1972, when it became illegal curriculum that would cause a school to lose its Federal funding, tax breaks and education status. Back in my day as a mathematician this symbol " / " was a fraction symbol it meant to take only the value to the left and only divide it by the value to the right. This symbol " ÷ " the division symbol took everything to the left of it and divided it by everything to the right of it, up to the next division symbol. It was an inline way to do complex engineering and geometric equations. But I would not worry too much, I doubt we will ever fix it. Sincerely, William McCormick

It's theroretical so ofcourse it doesn't absolutely match, pi