The volume of a sphere is similar to that of a cylinder.  If you place a sphere with the same length of radius as the base of the cylinder inside a cylinder, it does not fill the cylinder.  Therefore, the volume of a sphere must be smaller than the volume of a cylinder with its base congruent to the great circle of the sphere.  The volume of a sphere is actually two-thirds that of the cylinder.  The video below verifies this fact, please watch it prior to reading the rest of this page.

Now using the information above, let's transform the formula for the volume of a cylinder into the volume of a sphere.  The height of the cylinder is the length of the diameter of the sphere. 

Volume of Sphere



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