-Knowing That the Surface Area of a Sphere Is Given

Question

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The Answer is The volume of the sphere is the approximation of the sum of the volumes of the n pyramids.
Where the areas of the bases are

,

, and so on through

, the corresponding altitudes all have a length approximated by h. The the volume of the sphere is appproximated by

. In turn,

.
As n increases, the length h of the altitudes of the pyramids approaches r, the length of the radius of the sphere; that is,

. As n increases, the combined base areas of the pyramids also approaches the surface area S of the sphere; that is,

.
As n increases, we find that

or

, so that

.

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