Session 2     October 17, 2007

Dr. David Gay presented Understanding Volume Formulas by Using Geometry Models

Many people memorize mathematical formulas with little understanding of the components, which often leads to faulty memory of the formulas. David Gay used numerous polyhedra models to enable the participants to develop sense of how the formulas are derived.

Discussions began with area formulas of rectangles, triangles, circles and irregular polygons. This led to the derivation of volume formulas for prisms, cylinders, pyramids, and cones. To demonstrate a model for the surface area of a sphere, Dr. Gay tightly wrapped a sphere with clothesline. He used a second clothesline of the same length to create a circle. Th group compared the radius of the sphere, R, to the radius of the circle, 2R, to illustrate convincingly that the surface area of a sphere is equal to 4 pi R squared.

To develop a formula for the volume of a sphere, participants imagined that the sphere could be sliced into curvy-bottomed pyramids to the center of the sphere. The pyramids each have a volume equal to 1/3 the area of the base times the height. All pyramid volumes would be added to get the total volume of the sphere. Since the areas would sum to the surface area of the sphere (4 pi R squared) and the height of each pyramid is the radius of the sphere, the volume formula becomes: 1/3 times 4 pi R squared times R. This simplifies to 4/3 pi R cubed.

Dr. Gay also related Archimedes discovery of the ratio of the surface area of a sphere to the smallest cylinder able to contain it- 2 to 3. The ratio of the volumes of those same figures is also 2 to 3. As David Gay would say, Neat!

Participants received packets containing the presentation slides and templates for creating pyramids and prisms.