-Knowing that the surface area of a sphere is given by , complete the proof of the
volume formula for the sphere. Having separated the sphere into n pyramids ( each with the center of the sphere for its vertex), show that the volume of the sphere is .

Question

-Knowing that the surface area of a sphere is given by  , complete the proof of the
volume formula for the sphere. Having separated the sphere into n pyramids ( each with the center of the sphere for its vertex), show that the volume of the sphere is  .

Quizplus · Accepted Answer

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

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 .

,

, 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

.

-Knowing That the Surface Area of a Sphere Is Given