Sphere Mathimatics

**Q_Me** · Dec 6th, 2002, 03:01 PM

(4/3)(pi)(r^3) = Volume of a Sphere

right?

So why is (4/3) a factor? What is the significence of 4/3? I never really understood that.

I tried to prove that theory of sphere volume, by using areas of 2D cirlces on top of one another. However I did not know the formula for that. So how can you prove the sphere volume theory?

**bugzpodder** · Dec 6th, 2002, 03:55 PM

i think you have to use intergration (calculus) to prove it.

**Kalkewl8ter** · Dec 6th, 2002, 10:11 PM

Divide a sphere into an infinite amount of pyramid-like solids with a vertex at the centre and infinitely small bases on the surface of the sphere. Let the area of the base of one of these solids be B.
The volume of one of these solids is (1/3)*B*r, where r is the radius of the sphere, and also the height of one of the pyramid-like solids. Since the surface area of a sphere is 4*pi*(r^2), there are a total of (4*pi*(r^2))/B pyramid-like solids. Therefore, the total volume of the sphere is:

(1/3)*B*r*((4*pi*(r^2))/B)=(4/3)*pi*(r^3)

QED

**bugzpodder** · Dec 6th, 2002, 11:26 PM

knowing that you have some calculus knowledge Kalkewl8ter, if you have the function

f(r)=4pi(r^2)
if you intergrate it:

F(r)=I f(r)dr=(4/3)pi(r^3)

in other words, the derivitive of F(r) is f(r)

**Kalkewl8ter** · Dec 6th, 2002, 11:38 PM

That's cool, bugz!
Btw, can anyone explain why integrals are related to area, and derivatives are not, yet they are opposites of eachother?

**bugzpodder** · Dec 6th, 2002, 11:43 PM

its is just a geometric interpretation. as long as you know how to do intergrals, you are ok (just like as long as you know that 2+2=4 and not 5, you are ok)

**Kalkewl8ter** · Dec 7th, 2002, 10:29 PM

Unless of course so guy is standing over you with a pain generator...
Bugz, you are REALLY bitter at English!

**SilverSprite** · Dec 8th, 2002, 08:54 PM

Kalk, for your solution thingy. Are the bases of the pyramids with area B rounded? I dont see how to volume is the same if they are and I don't see how your thing makes sense if they aren't. Sorry i'm not as bright.

**Q_Me** · Dec 9th, 2002, 03:41 PM

I'm still confused here.

I can understand using pyramid volumes in a sphere, but what the hell did you just say?

Give me a better extended Equation with no Calculus or Geometry. And what the hell are Intergals?

**SilverSprite** · Dec 9th, 2002, 04:28 PM

Integrals are part of calculus used to do who knows what. Bugz and kalk are just trying to feel special.

**Q_Me** · Dec 10th, 2002, 03:37 PM

So let me get this right.

You have infinit amount of Pyramids with their bases on the outside of the sphere and their points at the center of the sphere.

You stack them in all 4 directions of them selves, their total volume of all pyramids is equal to the sphere's volume.

But still how would it be done? and what is the equation of a half circle; the equation of the parabola mapped on a 2D graph that is equal to a half circle?

Y = Sqrt (Diameter^2 - (X^2)) is the code for maping a half cirlce and doing it negative makes the complet circle. Right? So if you stack areas of circles on the Z Axis that matches the width from side to side of the parabola's x/y postitions. You should also get the area of a sphere.

I think I got it.

How's that for Algebra Kalk? A-l-g-e-b-r-a NOT CALCULUS!

**bugzpodder** · Dec 10th, 2002, 04:40 PM

Kalk and me ARE special, and so are you, sprite

**Q_Me** · Dec 11th, 2002, 11:24 AM

Originally posted by bugzpodder
Kalk and me ARE special, and so are you, sprite

Sprite? I'm not a three-dimensional object!

**Kalkewl8ter** · Dec 13th, 2002, 05:12 PM

Algrebra? I used algebra for my entire solution. It's bugz that notices these weird smart little things with calculus. And sprite, the bases don't have to be rounded, because they're so infinitely small that it doesn't matter. (lim B-->0) Just as with the essential idea of integral calculus, a surface with a curved perimeter can be split into many many rectangles with infinitesimally small width.

**bugzpodder** · Dec 13th, 2002, 05:55 PM

actually mrs romiens showed me that

**Q_Me** · Dec 16th, 2002, 03:28 PM

Who cares about the Romiens Integral Theory

**bugzpodder** · Dec 16th, 2002, 03:48 PM

just because you don't care doesnt mean other ppl don't care! WAY TO BE AN ASS!

**bugzpodder** · Dec 16th, 2002, 03:49 PM

Originally posted by Q_Me
Sprite? I'm not a three-dimensional object!

aint talking to you, buddy! talking to silversprite

**MrPolite** · Dec 16th, 2002, 11:04 PM

Originally posted by bugzpodder
its is just a geometric interpretation. as long as you know how to do intergrals, you are ok (just like as long as you know that 2+2=4 and not 5, you are ok)

you dont have to just accept the definition of integrals

integral is just another way of writing that stupid sumation notation.... I dont remember it now

but to get the volume, you write the volume as a sumation formula and you can then convert that to integral

**Kalkewl8ter** · Dec 17th, 2002, 05:33 PM

Hey, calm down people! I wonder how mathematicians ever meet in person without massacring eachother.

Mrs Romiens is not as bad as I at first thought anyways, though, bugz, but I still hate the way she does math.

**SilverSprite** · Dec 17th, 2002, 05:36 PM

I dont like her

makes my blood boil

**bugzpodder** · Dec 17th, 2002, 06:02 PM

i didn't ask you ppl if you like mrs romiens or not to me shes just great she taught me calculus and i actually understood it with only grade 9 knowledge so anyone who says she can't teach is obviously wrong i would never buy that capache

**SilverSprite** · Dec 17th, 2002, 06:28 PM

I didn't say she can't teach. I think she teaches well, but I believe that she is discriminating against some of us.

**bugzpodder** · Dec 17th, 2002, 07:08 PM

oh ur bro said she sucks at teaching and no one could understand her except for him and a few others

**SilverSprite** · Dec 17th, 2002, 07:22 PM

I'm not my bro now am I?

**bugzpodder** · Dec 17th, 2002, 07:25 PM

no one said you were. i just don't understand why mrs romiens would some ppl's blood boil.

**SilverSprite** · Dec 17th, 2002, 08:30 PM

Umm how come you can't understand that she doesnt like everyone she meets. I dunno, she's nice to me. She just thinks i'm stupid.

**bugzpodder** · Dec 17th, 2002, 10:32 PM

she gave you an A+ in math that means she thinks you are pretty smart at least in her class. if she thinks you are stupid the highest mark you are going to get is a 70.

**Kalkewl8ter** · Dec 20th, 2002, 04:04 PM

Romiens can teach, just not with the winding function. I'm not complaining about marks anymore, just her love of SOHCAHTOA.

**bugzpodder** · Dec 20th, 2002, 05:00 PM

i use doublethink: the ability to hold two contradictary thoughts at one time and believe in both of them. I love SOHCAHTOA and I hate SOHCAHTOA.

**Kalkewl8ter** · Dec 20th, 2002, 05:02 PM

Doublethink and math do not mix.

**bugzpodder** · Dec 20th, 2002, 05:05 PM

they just did

**Guv** · Dec 25th, 2002, 05:28 PM

Kalkewl8ter: Just noticed your smoke and mirrors post. Very cute, but how do you know that the surface area of a sphere is 4*Pi*r²?

Does anybody know of a geometric or algebraic derivation of either formula? The proof using integrals is not difficult if you know calculus, but otherwise it is incomprehensible.

**bugzpodder** · Dec 25th, 2002, 10:42 PM

I know that Archimedes is the first to derive these equations, but i have never seen the proof. you may find it if you do a search

**bugzpodder** · Dec 25th, 2002, 10:57 PM

the chinese have developed a truely remarkable proof, which can be found here: http://www.staff.hum.ku.dk/dbwagner/...SPHERE-TX.html

**bugzpodder** · Dec 25th, 2002, 11:06 PM

Here is Archimedes' work on spheres, notice that they had taken "advantage of some trigonometry and calculus notation to simplify the derivation", which probably was not what you had in mind: http://www.mathpages.com/home/kmath343.htm

**Jared** · Dec 28th, 2002, 03:44 PM

You know, I was reading this thread and I thought, Hey, this thread needs yet another reply.

In simple terms, because I am a simpleton, this is the significance of the 4/3 in the formula for volume of a sphere.

First cut the Sphere in half, (with a sharp knife) blah blah, the height is now equal to the radius. So from here on out, when you see an r^3, I have skipped the step of calling the height h and just combined it with the radius r.

Okay, here is the formula for volume of a sphere = 4/3 * pi * r^3

Working backwords, the 4/3 is actually 2 * (2/3 * pi * r^3).

The (2/3 * pi * r^3) is the volume of a hemisphere, which is, as you know, half of the sphere.

So where the hell did the 2/3 come from? That was probably your next question, right?

Well the area of the hemisphere is the volume of a cylinder - the volume of a cone. so....

Volume of a cylinder = pi * r^3 , it is pi * r^2 * h but we are combining it to r^3, since the height is the same as r.
Volume of a cone = 1/3 * pi * r^3.

so... (pi * r^3) - (1/3 * pi * r^3) = 2/3 * pi * r^3, and if you times that by 2, you get 4/3!!!!

Is that the answer you were looking for??

**bugzpodder** · Dec 28th, 2002, 05:07 PM

why don't derive what you said! we are more interested in proofs than answers.

Well the area of the hemisphere is the volume of a cylinder - the volume of a cone. so....

prove that the volume of the hemisphere of radius r is the difference between the volume of a cylinder of base radius r and the volume of a cone of base radius r!

**Jared** · Dec 28th, 2002, 05:57 PM

Well, I thought the question was:

So why is (4/3) a factor? What is the significence of 4/3? I never really understood that.

So, I answered the 4/3 issue.

As for why the volume of a hemisphere is equal to the volume(cylinder) minus the volume(cone) question...

I shall close my eyes an hummmmmmmmm, and try to summon Euclid...

Euclid said that any cone is a third part of the cylinder which has the same base with it and equal height. And doubling this gives our sphere. Euclid proved this by proving that a cone could not be more than 1/3, and proved that it could not be less than 1/3.

Not the proofs you were looking for, I know, but who can spit that proof out??? (with out a little refreshing)???

After I take a look at it again, I'll attempt to do the proof.

Until then,

**bugzpodder** · Dec 28th, 2002, 06:50 PM

Archimedes first derived these formulas.

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