GCSEC: Non Round Shapes

NON ROUND SHAPES

Find the board with the four two-dimensional constant-width figures in white plastic. Between them is a square whose sides are 3.5". Place the circle in the square and notice that the circle always touches each of the sides of the square. Now place one of the other figures in the square. Slowly rotate the figure and notice that it is always touching all four sides of the square.

Put into the square the "triangle" shape with the sharp corners. As you rotate this figure watch what shape or pattern the three corners trace out. Continue rotating this figure and watch what shape or pattern the handle traces out.

Each of the figures is always in contact with all four sides of the square, even while being turned in either direction. This shows that when used as rollers they do in fact have a constant width.

The corners of the "triangle" trace out the outside edge or perimeter of the square. If the corners of this figure were cutting edges (like on a drill bit) you would be "drilling" a square hole.

A Pennsylvania engineer designed in the early 1900's a "drill" and chuck that would cut square holes. The primary anticipated use was to make square holes in iron and other metals. Watts Brothers Tool Works still makes equipment that can drill squares, hexagons and octagons. Today their square drills are used extensively by furniture manufacturers.

The descriptions for making constant width figures are included in the label for the NON ROUND ROLLERS exhibit. You can easily cut these figures and a corresponding square from cardboard or Styrofoam.

How could you compare the volumes of the different shapes? How could you calculate the surface area of the bottom of each shape?

The easiest way to compare the volumes would be using a plastic or glass container like a kitchen measuring glass that has the volume marked on it. The measuring glass should be large enough that any of the shapes could be completely immersed. If the object is simply to compare the volumes, the handles on the shapes would not have to be removed, since they are all identical. Place any one of the shapes in the measuring glass, cover it with water, read the volume, remove the shape and read the new volume.

If the object is to calculate the surface area of the bottom, remove the handle from the object. Find the volume as above. Convert the volume to cubic inches. Divide this volume by .75 because the shapes are 3.4" thick, and the result is the surface area in square inches.