BOOK VII. POLYEDRONS. 1. DEFINITION. A polyedron is a geometrical solid bounded by planes. The bounding planes, by their mutual intersections, limit each other, and determine the faces (which are polygons), the edges, and the vertices, of the polyedron. A diagonal of a polyedron is a straight line joining any two of its vertices not in the same face. The least number of planes that can form a polyedral angle is three; but the space within the angle is indefinite in extent, and it requires a fourth plane to enclose a finite portion of space, or to form a solid; hence, the least number of planes that can form a polyedron is four. 2. Definition. A polyedron of four faces is called a tetraedron ; one of six faces, a hexaedron; one of eight faces, an octaedron; one of twelve faces, a dodecaedron; one of twenty faces, an icosaedron. 3. Definition. A polyedron is convex when the section, formed by any plane intersecting it, is a convex polygon. All the polyedrons treated of in this work will be understood to be convex. 4. Definition. The volume of any polyedron is the numerical measure of its magnitude, referred to some other polyedron as the unit. The polyedron adopted as the unit is called the unit of volume. To measure the volume of a polyedron is, then, to find its ratio to the unit of volume. 5. Definition. Equivalent solids are those which have equal volumes. PRISMS AND PARALLELOPIPEDS. 6. Definitions. A prism is a polyedron two of whose faces are equal polygons lying in parallel planes and having their homologous sides parallel, the other faces being parallelograms formed by the intersections of planes passed through the homologous sides of the equal polygons. The parallel faces are called the bases of the prism; the parallelograms taken together constitute its lateral or convex surface; the intersections of the lateral faces are its lateral edges. The altitude of a prism is the perpendicular distance between the planes of its bases. A triangular prism is one whose base is a triangle; a quadrangular prism, one whose base is a quadrilateral; etc. 7. Definitions. A right prism is one whose lateral edges are perpendicular to the planes of its bases. In a right prism, any lateral edge is equal to the altitude. An oblique prism is one whose lateral edges are oblique to the planes of its bases. In an oblique prism, a lateral edge is greater than the altitude. 8. Definition. A regular prism is a right prism whose bases are regular polygons. 9. Definition. If a prism, ABCDE-F, is intersected by a plane GK, not parallel to its base, the portion of the prism included between the base and this plane, namely ABCDE-GHIKL, is called a truncated prism. F K B 10. Definition. If a plane intersects a prism at right angles to its lateral edges, the section is called a right section of the prism. 11. Definition. A parallelopiped is a prism whose bases are parallelograms. It is therefore a polyedron all of whose faces are parallelograms. From this definition and (VI. 32) it is evident that any two opposite faces of a parallelopiped are equal parallelograms. 12. Definition. A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to the planes of its bases. Hence, by (VI. 6), its lateral faces are rectangles; but its bases may be either rhomboids or rectangles. A rectangular parallelopiped is a right parallelopiped whose bases are rectangles. Hence it is a parallelopiped all of whose faces are rectangles. Since the perspective of figures in space distorts the angles, the diagram may represent either a right, or a rectangular, parallelopiped. 13. Definition. A cube is a rectangular parallelopiped whose six faces are all squares. PROPOSITION I.-THEOREM. 14. The sections of a prism made by parallel planes are equal polygons. Let the prism AD' be intersected by the parallel planes GK, G'K'; then, the sections, GHIKL, G'H'I'K'L', are equal polygons. For, the sides of these polygons are parallel, each to each, as for example, GH and G'H', being the intersections of parallel planes with a third plane (VI. 25), and they are equal, being parallels included between parallels (I. 104); hence, also, the angles of the polygons are equal, each to Ꭺ G' E' A' D B C K' H K G H E B each (VI. 32). Therefore, the two sections, being both mutually equilateral and mutually equiangular, are equal. 15. Corollary. Any section of a prism, made by a plane parallel to the base, is equal to the base. PROPOSITION II.-THEOREM. E A' D 16. The lateral area of a prism is equal to the product of the perimeter of a right section of the prism by a lateral edge. Let AD' be a prism, and GHIKL a right section of it; then, the area of the convex surface of the prism is equal to the perimeter GHIKL multiplied by a lateral edge AA'. B C H K E A D For, the sides of the section GHIKL being perpendicular to the lateral edges AA', BB', etc., are the altitudes of the parallelograms which form the convex surface of the prism, if we take as the bases of these parallelograms the lateral edges, AA' BB', etc., which are all equal. Hence, the area of the sum of these parallelograms is (IV. 10), GH × AA' + HI × BB' + etc. =(GH+HI+ etc.) X AA'. B 17. Corollary. The lateral area of a right prism is equal to the product of the perimeter of its base by its altitude. PROPOSITION III.-THEOREM. 18. The four diagonals of a parallelopiped bisect each other. Let ABCD-G be a parallelopiped; its four diagonals, AG, EC, BH, DF, bisect each other. Through the opposite and parallel edges AE, CG, pass a plane which intersects the parallel faces ABCD, EFGH, in the parallel lines AC and EG. The figure ACGE is a parallelogram, and its diagonals AG and EC bisect each other in the point 0. In the same manner it is shown that AG and BH, AG and DF, bisect each other; therefore, the four diagonals bisect each other in the point 0. D H A G B F 198 200 GEOMETRY is called the centre of the parallelopiped; and it is easily proved that 19. Scholium. The point 0, in which the four diagonals intersect, any straight line drawn through O and terminated by two opposite faces of the parallelopiped is bisected in that point. PROPOSITION IV.-THEOREM. 20. The sum of the squares of the four diagonals of a parallelopipe is equal to the sum of the squares of its twelve edges. In the parallelogram ACGE we have (III. 64), 2 AG2+ CE2 = 2AЕ2 + 2AC2, and in the parallelogram DBFH, BH2 + DF2 = 2BF2 + 2BD2. Adding, and observing that BF AE, and also that in the parallelogram ABCD, 2AC+2BD': 4AB4AD, we have D H G 4AE2 + 4AB2 + 4AD3, 21. Corollary. In a rectangular parallelopiped, the four diagonals are equal to each other; and the square of a diagonal is equal to the sum of the squares of the three edges which meet at a common vertex. Thus, if AG is a rectangular parallelopiped, we have, by dividing the preceding equation by 4, AG2 = AE2 + AB2 + AD2. 22. Scholium. If any three straight lines AB, AE, AD, not in the same plane, are given, meeting in a common point, a parallelopiped can be constructed upon them. For, pass a plane through the extremity of each line parallel to the plane of the other two; these planes, together with the planes of the given lines, determine the parallelopiped. In a rectangular parallelopiped, if the plane of two of the three edges which meet at a common vertex is taken as a base, the third edge is the altitude. These three edges, or the three perpendicular |