Technical Mathematics, Sixth Edition

6.5. Polyhedra

By geometric solid we mean a closed surface in space. Sometimes the word solid is taken to mean the surface, and sometimes the surface itself together with its interior. Keep in mind that we are talking about geometric figures and not real objects, and the word solid should not be taken here in its usual sense to imply rigidity, as in "solid as a rock." We will refer to a soap bubble, for instance, as a spherical solid even though the soap film and the enclosed air are far from rigid.

6.5.1. Volume and Area of a Solid

The volume of a solid is a measure of the space it occupies or encloses. A cube of side 1 unit has a volume of 1 cubic unit, and we can think of the volume of any solid as the number of such cubes it contains. This may not be a whole number or even a rational number, as we will see with the volumes of cylinders and spheres.

We will speak about three different kinds of areas in connection with solids: (a) the surface area will mean the total area of the solid, including any ends; (b) the lateral area which does not include the area of the ends or base(s), which we will define for each solid; and (c) the cross-sectional area, which is obtained when a solid is sliced in a certain way.

6.5.2. Polyhedra

We saw earlier that a polygon is a plane figure bounded by line segments. Now we define a polyhedron as a solid bounded by polygons, now called faces. Two faces meet in an edge, and the point where three or more edges meet is called a vertex (Fig. 6-89 ...