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 Question from Ron: I have read about Archimedes and his work with sphere in cylinder and cone in cylinder and the volume relationships. Did he or any others also extend this to regular based polygon based regular like pillars, and columns? The ratio of 1/3 to 1 whole holds true with all regular based columns as example: a regular pyramid having a regular hexagon base inside a regular hexagon column of equal height.

Ron,

By coincidence, an article appeared in the most recent issue of THE COLLEGE MATHEMATICS JOURNAL, 46:3 (May 2015), pages 162-171, that clarifies what Archimedes did concerning volumes and areas, and what results were known long before him; the article is “Circular Reasoning: Who First Proved That C Divided by d Is a Constant?” by David Richeson. Most of the results you mention are in Euclid’s ELEMENTS, and were probably discovered much earlier. There you learn in Book 12 that

• the area of a circle divided by the square of its radius is constant for all circles,
• the volumes of cones and pyramids are 1/3 the volume of the corresponding cylinder,
• the volume of a sphere is proportional to the cube of its radius.

Euclid did not mention anything about the circumference of the circle, or of the surface area of a sphere.

One of the many remarkable achievements of Archimedes (who came a generation or so after Euclid) was to prove that the same constant which we call pi equals

• the circumference of a circle divided by its diameter,
• the area of a circle divided by the square of its radius,
• (3/4) times the volume of a sphere divided by the cube of its radius,
• (1/4) times the surface area of a sphere divided by the square if its radius.

Of course, the volume of a cylinder is (3/2) times the volume of the sphere of the same radius (that is, the same as the radius of the base of the cylinder, while the height of the cylinder equals the diameter of the sphere), while the surface area of a cylinder (including its top and bottom) is also (3/2) times the surface area of the sphere with the same radius. According to Plutarch, Archimedes was so proud of his three-halves theorem, that he requested it be inscribed on his tomb.

Chris

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