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Measurements in Three Dimensions

In Geometry 1, we were introduced to the idea of three-dimensional surfaces. We chiefly studied simple closed surfaces and, more specifically, polyhedrons. Remember, polyhedrons are surfaces made up entirely of polygons. The surfaces we studied rarely exist alone in the world--they are usually united with their interior points to form a three-dimensional solid, like a ball of clay, for example. Three-dimensional solids have measurements analogous to perimeter and area; they are called surface area and volume. Where perimeter is a measure of length only--it is a one- dimensional measure for figures of two dimensions--surface area is a measure solely of area, a two-dimensional measure of solids that exist in three dimensions. Both surfaces and solids have surface area. The surface area of a solid is simply the area of the surface that encloses it.

Solids also have volume, the three-dimensional equivalent of area. The most prevalent way to compare solids is by their volume. In the following lessons we'll discuss the volume of such surfaces as cylinders, cones, and spheres. In reality, these surfaces have no volume because they are two-dimensional, but here we'll refer to the solids which they bound as the surfaces themselves. For example, we'll call the solid bound by a prism a prism, the solid bound by a cone a cone. This way, when we learn about volume, we don't need to keep saying, "the volume of the solid bound by a..."

The reason for this lengthy explanation is that keeping track of dimension is one of the most important tasks of a geometry student, and you should never fall into the trap of thinking that certain objects have more dimensions than they really have. So remember that surfaces like prisms and pyramids are two-dimensional, even though in this section we'll use their names to denote the solids that they bound in an effort to explain volume without extra language.