Archimedes and Cavalieri's Principle

Dr. Sarah's MAT 3610: Introduction to Geometry
Archimedes is considered to be one of the greatest mathematicians and scientists. According to Plutarch (AD 45-120), Parallel Lives: Marcellus, Archimedes had requested that a pictorial representation of a sphere and a cylinder appear on his tombstone. From this, we can infer that he must have considered his work on a sphere and a cylinder to be one of his greatest accomplishments. Cicero (106-43 BC), in Tusculan Disputations, Book V, Sections 64-66, states that he went to Syracuse and indeed found the grave which contained the pictorial representation along with text verses.

The formulas for the volume and surface area of a cylinder were known before Archimedes' time, but those for a sphere were not known. Archimedes wanted to find exact expressions for the volume and surface area of a sphere, and he did indeed do just this by using ideas related to Cavalieri's Principle.
  1. Fill up the sphere with sand and pour it into the cylinder. Eyeball this: approximately what fraction of the cylinder does the sphere take up?

  2. How many cones of sand does it take to fill up the cylinder?

  3. What fraction of the cylinder does the cone take up?

  4. Use only your answers in 1. and 2. to make (and write down) a conjecture relating the cylinder to the cone plus the sphere.

  5. Test your conjecture with sand and explain your results.

  6. Sketch the figures and label all the dimensions in terms of r, the radius of the sphere, including the height of the cone and cylinder as a function of r.

  7. Next, test your conjecture by using formulas for the volume of these three objects, in terms of r.

  8. Archimedes was trying to derive the formula for the volume of a sphere, so he could not assume this formula anywhere in his work. I've found some interactive web pages that will give you an idea of his methods, but they are slightly different than his original construction, which is more accurately set up above. Instead of a sphere of radius r, these explorations begin with half a sphere (or hemisphere) of radius r, and a cylinder and cone that are half as tall:   and on   to see Archimedes' argument.
    What familar theorems are assumed in the proof?