# How do you find the area of a sphere based on it's volume?

1. May 25, 2005

### Serj

How do you find the area of a sphere based on it's volume?

2. May 25, 2005

### JamesU

surface area of sphere: 4pi r^2

3. May 25, 2005

### whozum

Well, usually you find both of the above based on the circle.
Integrating half a circle's arc length gives surface area, and integrating half a circle's area gives volume.

However, the sphere's volume [itex] \frac{4}{3}\pi r^3 [/tex], and surface area [itex] 4\pi r^2 [/tex] are related as integrals/derivatives of the radius 'r'.

Last edited: May 25, 2005
4. May 25, 2005

### Serj

Can you put that in an equation form like Area= volume x _ ?

5. May 25, 2005

### JamesU

You mean turn the volume into the SA?

6. May 25, 2005

### Serj

Yes .

7. May 25, 2005

### whozum

Well I guess a really ugly way of doing it is just saying

$$SA = \frac{3V}{r}$$

Do you know calculus?

8. May 25, 2005

### Serj

No, i just finished algebra II. Is there another way to say it? Without having to first calculate radius?

9. May 25, 2005

### Hurkyl

Staff Emeritus
You have all you need to figure that out yourself...

10. May 25, 2005

### arildno

Serj:
Do you agree that the radius r in terms of volum V is: $$r=(\frac{3V}{4\pi})^{\frac{1}{3}}$$??
Use this to express the surface area in terms of the volume.

11. May 25, 2005

### dextercioby

Serj,the sphere's volume ("it's (sic!) volume") is zero...So the area is simply

$$\mbox{Area}_{\mbox{Sphere}} =\mbox{Area}_{\mbox{Sphere}} \ + 0$$

$$\Longrightarrow \mbox{Area}_{\mbox{Sphere}} =\mbox{Area}_{\mbox{Sphere}} \ + \ \mbox{Volume}_{\mbox{Sphere}}$$

Daniel.

P.S.Apart from the notation,there's no joke in this post... :uhh:

12. May 25, 2005

### arildno

Daniel is of course right.

You are to express the sphere's area in terms of the enclosed ball's volume.

Last edited: May 25, 2005
13. May 25, 2005

### Curious3141

I could see that one coming a mile away. LOL !! :D

14. May 25, 2005

### chroot

Staff Emeritus
$$A = (36 \pi V^2)^{1/3}$$

- Warren

15. May 25, 2005

### mathwonk

to find the volume of a sphere one computes how the volume form the portion of the sphere at or below height y, changes as y changes. it turns out that the rate of change of the volume at height y, equals the area of the circular slice at height y.

from this one can write a formula for the dertivative of the rising voilume formula, and from that one can write a formula for the volume of a sphere of radius R as
(4/3)pi R^3).

then one writes the volume of the sphere in terms of the radius, and sees that the rate of change in that case, at the point where the radius is r, equals the area of the sphere of radius r. Since we know the volume, and that the area is the derivative of the volume wrt the radius, we get that the area is 4 pi R^2.

so first you find the volume of the sphere, knowing the area of a circle. Then knowing the volume of a sphere you find the area of the sphere.

I am not sure, since I have not seen his works, but i suspect this is the way Archimedes did it.

16. May 26, 2005

### Galileo

:rofl:

I was waiting for the same thing to happen too.

17. May 26, 2005

### Serj

Thanks, that's just what I was looking for.