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what is the volume of a sphere?

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what is the volume of a sphere?

Dec 8, 2014

#3
+98173
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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

Dec 8, 2014

#1
+270
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Volume of the sphere = 4/3ㅠr^3

Where, r = radius of the sphere

Derivation for Volume of the Sphere
The differential element shown in the figure is cylindrical with radius x and altitude dy. The volume of cylindrical element is...

The sum of the cylindrical elements from 0 to r is a hemisphere, twice the hemisphere will give the volume of the sphere. Thus,

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

Dec 8, 2014
#2
+17746
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The formula for the volume of a sphere is:  V  =  (4/3)·π·r³

Dec 8, 2014
#3
+98173
+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
#4
+99352
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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.

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.

Dec 8, 2014
#5
+98173
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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

Dec 9, 2014
#6
+99352
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Thanks chris - that is really neat :)

Dec 9, 2014