# Geometrical Concepts from Constructions, Models, and Investigations Summary

## Monday, July 8, 2002

Monday's discussion centered around a phenomena observed at lunch. An empty paper cup had been left on a table and was being pushed by a breeze. It rolled across the table, tracing out an annulus. Jim King brought it back to the group as a jumping off point for discussion. By using similar triangles, Steve showed how to find the slant height of the cone, of which the cup was a frustum. This height would be the outer radius of the annulus. The group then confirmed this by direct measurement. Cups with different upper and lower diameters, and different heights were also investigated. By observation and calculation the number of revolutions for the cup to trace the complete annulus was determined. This was shown to be dependent on the inner and outer diameters, and the height of the cup. A physical representation was also generated, by cutting the cup open and actually constructing the annulus.

Jim then referenced a puzzle from the Taggart Brothers of "Car Talk" fame. A father asks his son to measure the deck of a merry-go-round for them to determine the amount of paint they will need to paint it. The son returns with only one measurement, the length of a chord of the outer circle, tangent to the inner circle. His older brother arrives and announces that this single measurement is indeed sufficient. Not only did our group determine the area of the annulus, but Troy also demonstrated that this same chord and the diameter of the outer circle which intersects the chord at the point of tangency, form the basis for an elegant proof of the Pythagorean theorem. At this point we adjourned to the lab and used Geometer's Sketchpad 4 to construct a dynamic representation of the Car Talk problem.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.