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Volume of a sphere
By Danish Dewan on Sunday, March 17, 2002 - 07:03 pm:

How can I find the volume of a sphere? Can you explain the method clearly please?

By Julian Pulman on Sunday, March 17, 2002 - 08:44 pm:

The equation of a circle is y2 + x2 = r2, where r is the radius.

So, at anypoint on the circle, and area of a disc enclosed by the circle is py2 = p(r2 - x2).
Thus, summing up the discs of varying radii (corresponding to the equation), we integrate:

integrating

By Emma McCaughan on Monday, March 18, 2002 - 03:36 pm:

You didn't say how old you are, Danish. Having looked up what school you are at, I suspect that you don't know what integrate means.
The idea used above is one of splitting the sphere into lots and lots of slices.

I expect you will be able to use the formula given at the end (r is the radius of the sphere), but if you want to understand how the formula comes about, there is an explanation of how to find the volume of a sphere which you should be able to follow in two articles on the NRICH site:
Volume of a pyramid and a cone
Mouhefanggai

These articles use the slicing idea, but do it in a way which means you don't have to have studied A-level maths!

By Danish Dewan on Saturday, March 23, 2002 - 10:29 pm:

I'm only eleven but I know quite a lot for my age. I'm OK with the formula for V.
Thanks

By Kerwin Hui on Sunday, March 24, 2002 - 02:20 am:

Danish,

Do you understand why the formula of a cone is p/3×radius2×height? If so, here is a treatment of the volume of sphere (due to Archimedes).

Suppose you are given a right cone and a sphere, both of radius R (a right cone is one which radius=height). If you take a slice through the sphere at a distance d (0£d£R) above the equator, the radius of the resulting circle is given by r2=R2-d2. Also, if you take a slice through the cone at a distance d from its vertex, then the radius of circle is just d. So the area of the two circle adds up to a circle of radius R. Then, "summing up the slices", you get

½×Volume of sphere + Volume of right cone = Volume of cylinder

Substituting what we know:

½×Volume of sphere of radius R + (1/3)pR3=pR3

and now just rearrange to give the volume of sphere is 4/3×p×radius3.

Kerwin