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A solid is an enclosed portion of space bounded by plane and curved surfaces. Thus, the pyramid , the cube , the cone , the cylinder , and the sphere are solids. A solid has three dimensions: length, width, and thickness. Practical illustrations of solids include a box, a brick, a block, and a ball. These are not, however, the pure or ideal solids that are the concern of solid geometry. In solid geometry we study the geometric properties of perfect solids, such as their shape, their size, the relationships of their parts, and the relationships along solids; physical properties such as their color, weight, or smoothness are disregarded.



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Polyhedra A polyhedron is a solid bounded by plane (flat) surfaces only. Thus, the pyramid and cube are polyhedrons. The cone, cylinder, and sphere are not polyhedra, since each has a curved surface. The bounding surfaces of the polyhedron are its faces; the lines of intersection of the faces are its edges, and the points of intersection of its edges are its vertices. A diagonal of a polyhedron joins two vertices not in the same face. Thus, the polyhedron shown in Fig. 17-1 has six faces. Two of them are triangles (ABC and CFG), and the other four are quadrilaterals. Note AF is a diagonal of the polyhedron. The shaded polygon HJKL is a section of the polyhedron formed by the intersection of the solid and a plane passing through it.

Fig. 17-1





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The angle between any two intersecting faces is a dihedral angle. The angle between the covers of an open book is a dihedral angle. As the book is opened wider, the dihedral angle grows from one that is acute to one that is right, obtuse, and then straight. The dihedral angle can be measured by measuring the plane angle between two lines, one in each face, and in a plane perpendicular to the intersection between the faces.

A prism (Fig. 17-2) is a polyhedron two of whose faces are parallel polygons, and whose remaining faces are parallelograms. The bases of a prism are its parallel polygons. These may have any number of sides. The lateral (sides) faces are the parallelograms. The distance between the two bases is h; it is measured along a line at right angles to both bases. A right prism is a prism whose lateral faces are rectangles. The distance h is the height of any of the lateral faces.

Fig. 17-2

Fig. 17-3

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A rectangular solid (box) is a prism bounded by six rectangles. The solid can be formed from a pattern of six rectangles, as shown in Fig. 17-3, folded along the dashed lines. The length l, width w, and height h are its dimensions. A cube is a rectangular solid bounded by six squares. The cube can be formed from a pattern of six squares, as shown in Fig. 17-4, folded along the dashed lines. Each equal dimension is represented by e in the diagram.

Fig. 17-4

Fig. 17-5

A cubic unit is a cube whose edge measures 1 unit. Thus, a cubic inch is a cube whose edge is 1 in long. A parallelepiped is a prism bounded by six parallelograms (Fig. 17-5). Hence, the rectangular solid and cube are special parallelepipeds. The table that follows shows some relationships among polygons in plane geometry and the corresponding relationships among polyhedra in solid geometry. Plane Geometry Solid Geometry Polyhedron Polygon Quadrilateral Prism Parallelogram Parallelepiped Rectangle Square Rectangular solid Cube Pyramids A pyramid is a polyhedron whose base is a polygon and whose other faces meet at a point, its vertex. The base (B in Fig. 17-6) may have any number of sides. However, the other faces must be triangles. The distance from the vertex to the base is equal to the measure of the altitude or height h, a line from the vertex at a right angle to the base.

B Regular Pyramid (square base)

Fig. 17-6

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