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what is the volume of a sphere?

Guest Dec 8, 2014

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 #3
avatar+78643 
+10

Archimedes provided a rather ingenious proof of this by noting that the volume of a hemisphere is equal to the sum of the differences in the cross-sections of a cylinder and an associated cone both having the same radius as the hemisphere, i.e....... pi*r^2h - (1/3)pi*r^2h = (2/3)pi*r^2h. But, the cylinder and the cone also have a height equal to this radius, so we have (2/3)pi*r^2h = (2/3)pi*r^2*r = (2/3)pi*r^3

So....twice this equals the volume of a whole sphere = 4/3pi*r^3

 

CPhill  Dec 8, 2014
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 #1
avatar+270 
+10

Volume of the sphere = 4/3ㅠr^3

 

Where, r = radius of the sphere
 

Derivation for Volume of the Sphere
Figure for the Derivation of Formula of Sphere by IntegrationThe differential element shown in the figure is cylindrical with radius x and altitude dy. The volume of cylindrical element is...
$ dV = \pi x^2 dy $
 

The sum of the cylindrical elements from 0 to r is a hemisphere, twice the hemisphere will give the volume of the sphere. Thus,
$ \displaystyle V = 2\pi \int_0^r x^2 dy $

 
From the equation of the circle x2 + y2 = r2; x2 = r2 - y2.

$ \displaystyle V = 2\pi \int_0^r (r^2 - y^2) dy $

$ V = 2\pi \left[ r^2y - \dfrac{y^3}{3} \right]_0^r $

$ V = 2\pi \left[ \left(r^3 - \dfrac{r^3}{3}\right) - \left(0 - \dfrac{0^3}{3}\right) \right] $

$ V = 2\pi \left[ \dfrac{2r^3}{3} \right] $

$ V = \dfrac{4 \pi r^3}{3} $      

flflvm97  Dec 8, 2014
 #2
avatar+17655 
+5

The formula for the volume of a sphere is:  V  =  (4/3)·π·r³

geno3141  Dec 8, 2014
 #3
avatar+78643 
+10
Best Answer

Archimedes provided a rather ingenious proof of this by noting that the volume of a hemisphere is equal to the sum of the differences in the cross-sections of a cylinder and an associated cone both having the same radius as the hemisphere, i.e....... pi*r^2h - (1/3)pi*r^2h = (2/3)pi*r^2h. But, the cylinder and the cone also have a height equal to this radius, so we have (2/3)pi*r^2h = (2/3)pi*r^2*r = (2/3)pi*r^3

So....twice this equals the volume of a whole sphere = 4/3pi*r^3

 

CPhill  Dec 8, 2014
 #4
avatar+91038 
+5

Welcome to the forum fiflvm97.  I hope that ou like it here.      I can see that you will be a great asset to us if you choose to stay around :)

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I have been looking at these answers.

Geno, yours is concise but exactly what the asker would have wanted.    

 

flflvm,  I loved your analysis of this answer.  Adding the diagram made all the difference too.

I hope that some of our senior students really think about what you said - as I did.  It would be of great benefit to them.  

 

Chris,  I had not thought before about the volume of a cone being one third of a hemisphere.  It's an obvious conclusion now that you point it out.

As for the rest of what archimedes said that will require a more concentrated effort from me.  Thank you for discussing it.   

Melody  Dec 8, 2014
 #5
avatar+78643 
+5

Melody......there used to be a very good illustration of this on the net, but I'm unable to find it, now. However, this one is pretty good......http://mathcentral.uregina.ca/QQ/database/QQ.09.01/rahul1.html

 

CPhill  Dec 9, 2014
 #6
avatar+91038 
0

Thanks chris - that is really neat :)

Melody  Dec 9, 2014

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