Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Linear objects, like lines and segments, are one-dimensional. The only finite one-dimensional object is a line segment which is bounded by two endpoints. Length is the measurement of the interior space of a line segment. The length is also the distance between the endpoints. In a linear space, all objects lie on a single line. we move only forward and backward along that line. There is no height or width.

Two-dimensional geometry is the study of objects lying in a plane. Two-dimensional objects include polygons, circles, and other figures. A closed two-dimenional figure will be bounded by linear and/or curved segments that separate the interior of the figure from the rest of the plane, which we refer to as the exterior of the figure. The measure of the interior of a figure is called its area and involves both length and width. This second dimension, width, is measured in a direction perpendicular to that of length. A plane will also include objects of smaller dimension, such as one-dimensional lines and zero-dimensional points. We cannot measure area of these smaller dimensional objects.

In three-dimensional geometry, we have length, width, and a third direction called depth (or height). The geometrical solids studied in this section including prisms, cylinders, cones, pyramids, and polyhedra are examples of three-dimensional solids. Each three-dimensional solid is bounded by surfaces which may include polygons, circles, and other two-dimensional objects. The measurement of the interior space of a geometrical solid is called volume.

How does the change in the length in one or more directions affect the size of a figure’s interior?