566. Cor. I. The volume of a truncated right triangular prism is equal to the product of its base by one third the sum of its lateral edges. 1. What is the direction of the lateral edges AD, BE, CF with reference to the base ABC? 2. How, then, do AD, BE, and CF compare with the altitudes of the three pyramids whose sum is equivalent to ABC-DEF? 3. To what is the volume of each of these pyramids equal? E A C B 4. To what, then, is the volume of ABC-DEF equal? 567. Cor. II. The volume of any truncated triangular prism is equal to the product of its right section by one third the sum of its lateral edges. 1. If GHK is a right section, to what is the volume of GHK-DEF equal? 2. To what is the volume of GHKABC equal? 3. To what, then, is the volume of ABC-DEF equal? H Α B E Ex. 780. What is the lateral area of a right prism whose altitude is 12 in. and the perimeter of whose base is 20 in.? Ex. 781. Find the ratio of two rectangular parallelopipeds, if their altitudes are each 7m and their bases 3m by 4m and 7m by 9m, respectively. Ex. 782. Find the ratio of two rectangular parallelopipeds, if their dimensions are 2dm, 4dm, 3dm, and 6dm, 7dm, 8dm, respectively. Ex. 783. What is the volume of a rectangular parallelopiped whose edges are 20.5m, 12.75m, and 8.6m, respectively? Ex. 784. The altitude of a regular hexagonal prism is 12 ft., and each side of its base is 10 ft. What is its volume? Ex. 785. What is the volume of a pyramid whose altitude is 18dm and whose base is a rectangle 10dm by 6dm? Ex. 786. What is the volume of a truncated right triangular prism, if each side of its base is 3 ft. and its edges are 3 ft., 4 ft., and 6 ft., respectively? Ex. 787. What is the lateral area of the frustum of a square pyramid whose slant height is 13m, each side of the lower base being 3.5m, and of the upper base 2m ? Proposition XIX 568. Form two tetrahedrons which have a trihedral angle of one equal to a trihedral angle of the other. Considering homologous faces of these trihedrals as bases of the tetrahedrons, how does the ratio of their volumes compare with the ratio of the products of their bases by their altitudes? (§ 561) How does the ratio of their bases compare with the product of the ratios of the homologous edges which include the basal face angles of the equal trihedrals? (§ 340) How does the ratio of the altitudes of the tetrahedrons compare with the ratio of the third edges of the equal trihedrals? (§ 299) From these equal ratios discover how the ratio of the volumes of the tetrahedrons compares with the ratio of the products of the three edges of the equal trihedral angles. Theorem. Tetrahedrons which have a trihedral angle of one equal to a trihedral angle of the other are to each other as the products of the edges of the equal trihedral angies. Data: Any two tetrahedrons, as Q-ABC and T-DEF, which have the trihedral angles Q and T equal. To prove Q-ABC: T-DEF Proof. Apply T-DEF to angles T and Q coincide. QA × QB × QC: TD × TEXTF. Q-ABC so that the equal trihedral Draw Co and FP perpendicular to the plane QAB. Then, their plane intersects QAB in QPO. Now, co and FP are the altitudes of the triangular pyramids C-QAB and F-QDE; .. § 561, C-QAB: F-QDE = QAB X CO: QDE × FP. QAB: QDE= QA X QB: QD × QE. But, § 340, (1) C-QAB : F-QDE = QA × QB × CO: QD × QE × FP. (2) Now, .. § 299, rt. A QOC and QPF are similar; CO: FP QC: QF. Substituting in (2), that is, Why? C-QAB: F-QDE = Q-ABC: T-DEF : = QA × QB × QC: TD X TE × TF. Therefore, etc. SIMILAR AND REGULAR POLYHEDRONS Q.E.D. 569. Polyhedrons which have their corresponding polyhedral angles equal, and have the same number of faces similar each to each, and similarly placed, are called Similar Polyhedrons. Faces, edges, angles, etc., which are similarly placed in similar polyhedrons are called homologous faces, edges, angles, etc. 570. Since the homologous sides of similar polygons are proportional, the homologous edges of similar polyhedrons are proportional. 571. Since similar polygons are proportional to the squares upon any of their homologous lines, the homologous faces of similar polyhedrons are proportional to the squares upon any of their homologous edges. 572. From § 571 it is evident that the entire surfaces of similar polyhedrons are proportional to the squares upon any of their homologous edges. Proposition XX 573. 1. Form two similar polyhedrons, and if possible divide them into the same number of tetrahedrons, similar each to each. 2. How does the ratio of any two homologous lines in two similar polyhedrons compare with the ratio of any two homologous edges? Theorem. Similar polyhedrons may be divided into the same number of tetrahedrons, similar each to each, and similarly placed. SOLID GEOMETRY. BOOK VIII. y two similar polyhedrons, as AJ and A'J'. that AJ and A'J' may be divided into the same numhedrons, similar each to each, and similarly placed. elect any trihedral angle in AJ, as B, and through the of its edges, as A, G, C, pass a plane. Also through gous points, A', G', C', pass a plane. 310, in the tetrahedrons B-AGC and B'-A'G'C', the BAG, BGC are similar to the faces B'A'C', B'A'G', to each. AG: A'G' = BG : B'G' = CG: C'G', AG: A'G' CG : C'G' = AC: A'C'. face ACG is similar to face A'C'G'. Why? logous faces of these tetrahedrons are similar. 00, the homologous trihedral angles of these tetraequal. 569, tetrahedrons B-AGC and B'-A'G'C' are similar. these similar tetrahedrons are removed from the simirons of which they are a part, the polyhedrons which continue to be similar; for the faces and polyhedral he original polyhedrons will be similarly modified. nuing to remove similar tetrahedrons from them, the yhedrons may be reduced to similar tetrahedrons, and en have been divided into the same number of tetramilar each to each, and similarly placed. e, etc. Q.E.D. Proposition XXI 575. Form two similar tetrahedrons. How do the trihedral angles at the vertices compare? Then, how does the ratio of the tetrahedrons compare with the product of the ratios of the homologous edges of the corresponding trihedral angles? (§ 568) How do the ratios of these edges compare with each other? Then, how does the ratio of the tetrahedrons compare with the ratio of the cubes of any two homologous edges? Theorem. Similar tetrahedrons are to each other as the cubes of their homologous edges. C E B Data: Any two similar tetrahedrons, as Q-ABC and T-DEF. To prove Q-ABC: T-DEF = QA3 : TD3 = etc. Proof. § 569, trihedral Q = trihedral T; .. § 568, Q-ABC : T-DEF = QA × QB × QC: TD × TE × TF, In like manner, the same may be proved for any two homologous edges. Therefore, etc. Q.E.D. |