So far we've only been studying geometric figures that exist in a plane. Now that we understand the basics of plane geometry, we can take a brief look into the world of three-dimensional figures and shapes. Such three-dimensional objects have length, width, and a new third dimension, depth; they are known as geometric solids. To understand a geometric solid, we study the surface that forms its boundary. Such a surface does not have volume, but the region it encloses, the geometric solid, does.
The variety of geometric solids in existence is limitless, so we have to restrict the kinds that we study. We will concern ourselves with geometric solids that are bound by polyhedrons and other simple surfaces. Polyhedrons are special kinds of surfaces that are bound by parts of intersecting planes: polygons.
As we study surfaces, you'll probably notice many similarities between surfaces and figures in plane geometry. Throughout geometry, a given geometric figure in a certain dimension often has a counterpart in other dimensions. A segment's relationship to a line is much like that between a polygon and a plane, and a polyhedron and space. The main difference between these pairs of geometric figures is which dimension they reach. If a concept is difficult to understand in a certain dimension, it might be helpful to think about that concept's counterpart in another dimension--probably a lower one--to try to understand it better. The greater the dimension, the harder things become to visualize, so simplification may come from reviewing lesser dimensions.
After polyhedrons are discussed in general, we'll introduce specific types of them, including prisms, pyramids, and regular polyhedrons. As we study these, we'll also see their counterparts in ciruclar form--surfaces with a similar shape that are partially bound by circles instead of polygons. Such surfaces include circular cylinders, cones, and spheres.
Like the previous topic of circles, in the following lessons geometric surfaces will be introduced and defined, but not explored in full. That will have to wait until the SparkNotes in Part 2 of Geometry, when the surfaces are united with their interiors to form geometric solids. Then we can look more closely at the applications of the properties and definitions we learn here. For now, we'll study the surfaces that form the boundaries of geometric solids, and their properties. It all starts when a third dimension is introduced, and different planes intersect.