About the cover image (from Chapter 10 in my Geometry course): Archimedes realized that the volume of a sphere equals the volume of a cylinder (of the same height and radius) minus the volume of a cone (of the same height and radius). The version of the proof illustrated here uses a pair of cones whose apexes meet in the middle instead of a single cone. (It is easy to show that a single cone and pair of cones with the same radius and half the height have the same volume.) Any horizontal slice across the sphere and the [cylinder minus cones] intersects equal areas: a reduced size circular disk, in one case, and an annulus (washer-shape) in the other. You could in principle make stacks of thin cards that make a sphere in one case and the cylinder minus cones shape on the other. Since the volumes of cylinders (area of base times height) and cones (1/3 of area of base times height) are already established (at the time in the course when this is introduced) doing the algebra (2/3 area of central cross section times diameter) results in the formula for the volume of a sphere (4/3 pi R cubed). In the Geogebra demo, from which this illustration was taken, you can move the intersection plane up and down and rotate the view to see it from different angles.
Posted on: Fri, 19 Dec 2014 18:22:59 +0000
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