Sphere Peeling: Volume
- Tim Brzezinski
Imagine a sphere being made up of infinitely many smaller spheres (spherical "shells"). Each one of these smaller spheres would be nestled inside all the others spheres (shells) that are larger than it, and would contain more spherical shells that are smaller than it. In the following investigation, we'll attempt to "peel a sphere", so to speak, in order to discover a means of calculating its volume. Enjoy the show! As you do, pay careful attention to what you observe. Be sure to answer the questions that follow.
Questions: 1) The solid made up by all those colorful discs closely, but not exactly, resembles what other solid? 2) What is the formula for the volume of this solid? 3) Note that the base of this solid is the same as the surface area of the original sphere. What is the formula for the base area of this solid? 4) Note that the height of this solid is the same as the radius of this original sphere. 5) How does the volume of the original sphere compare with the volume of the newer solid formed? 6) Given your response to (5), substitute the expression you wrote for (3) and the radius, "R", into the formula you wrote in (2). Simplify your expression. 7) Given what you now have now written for (6), What is the formula for the volume of a sphere?