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

In summary, the conversation discusses finding the area of a sphere based on its volume. The formula for surface area of a sphere is given in terms of the radius r. It is then shown that the volume and surface area are related through integrals and derivatives of the radius. The conversation also touches on the method that Archimedes may have used to find the surface area of a sphere. Finally, a formula for the surface area of a sphere in terms of its volume is provided as A = (36 \pi V^2)^{1/3}.
  • #1
Serj
94
0
How do you find the area of a sphere based on it's volume?
 
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  • #2
surface area of sphere: 4pi r^2
 
  • #3
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'.
 
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  • #4
Can you put that in an equation form like Area= volume x _ ?
 
  • #5
You mean turn the volume into the SA?
 
  • #6
yomamma said:
You mean turn the volume into the SA?

Yes .
 
  • #7
Well I guess a really ugly way of doing it is just saying

[tex] SA = \frac{3V}{r} [/tex]

Do you know calculus?
 
  • #8
No, i just finished algebra II. Is there another way to say it? Without having to first calculate radius?
 
  • #9
You have all you need to figure that out yourself...
 
  • #10
Serj:
Do you agree that the radius r in terms of volum V is: [tex]r=(\frac{3V}{4\pi})^{\frac{1}{3}}[/tex]??
Use this to express the surface area in terms of the volume.
 
  • #11
Serj,the sphere's volume ("it's (sic!) volume") is zero...So the area is simply

[tex]\mbox{Area}_{\mbox{Sphere}} =\mbox{Area}_{\mbox{Sphere}} \ + 0 [/tex]

[tex] \Longrightarrow \mbox{Area}_{\mbox{Sphere}} =\mbox{Area}_{\mbox{Sphere}} \ + \ \mbox{Volume}_{\mbox{Sphere}} [/tex]

Daniel.

P.S.Apart from the notation,there's no joke in this post... :rolleyes:
 
  • #12
Daniel is of course right.

You are to express the sphere's area in terms of the enclosed ball's volume.
 
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  • #13
I could see that one coming a mile away. LOL ! :D
 
  • #14
[tex]A = (36 \pi V^2)^{1/3}[/tex]

- Warren
 
  • #15
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
Curious3141 said:
I could see that one coming a mile away. LOL ! :D
:smile:

I was waiting for the same thing to happen too.
 
  • #17
chroot said:
[tex]A = (36 \pi V^2)^{1/3}[/tex]

- Warren

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

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

How do you calculate the radius of a sphere based on its volume?

To find the radius of a sphere based on its volume, you can use the formula V = (4/3) * π * r^3, where V is the volume and r is the radius. Rearranging the formula, we get r = (3V / 4π)^(1/3). Therefore, to find the radius, you would need to take the cube root of (3V / 4π).

How can you determine the area of a sphere if you only know its volume?

If you only know the volume of a sphere, you can use the formula A = 4 * π * r^2, where A is the surface area and r is the radius. As mentioned before, the radius can be found by taking the cube root of (3V / 4π). Once you have the radius, you can plug it into the formula to find the surface area of the sphere.

Is there a simpler way to find the area of a sphere based on its volume?

Yes, there is a simpler formula that relates the volume and surface area of a sphere. It is A = √(3Vπ), where A is the surface area and V is the volume. This formula can be derived from the previous formula A = 4 * π * r^2 by substituting the value of r from the formula for volume.

How do you convert between different units when calculating the area of a sphere?

To convert between different units when calculating the area of a sphere, you need to make sure that the units for both the volume and surface area are consistent. For example, if the volume is given in cubic meters, the surface area should be in square meters. You can use conversion factors or unit conversion calculators to convert between different units.

What is the relationship between the volume and area of a sphere?

The volume and area of a sphere are related through the formula A = √(3Vπ). This means that as the volume of a sphere increases, its surface area also increases. However, the rate at which the surface area increases is slower compared to the increase in volume. This is because the radius of the sphere also increases, leading to a larger surface area but not in the same proportion as the volume.

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