For the two pyramids have the same base ADC, and equal altitudes, since EB is to the plane ADC. § 537 Ax. 11 § 646 § 665 For the two pyramids have equal altitudes, since EB is parallel to the plane of their bases; § 537 and they have equivalent bases, because the A DFC and AFC have the same base CF and equivalent altitudes, since AD is || to CF. ... ABC-DEF ... R = R'. § 605 Ax. 11 P' + Q'+R'≈ P+Q+ R. Q. E. D. 677 COROLLARY 1. 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. For the lateral edges AD, BE, and CF are to the base ABC (§ 602). .. they are the altitudes of the three pyramids whose sum is equivalent to the truncated prism (§ 676). Denote the volume of the truncated prism by V, and its base ABC by B. B E F C Then V=B × AD + B × §BE + B × CF = B × }(AD + BE + CF). 678 COROLLARY 2. 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. For the right section DEF divides the truncated prism into two truncated right prisms the volumes of which are K G H D F Adding, volume of ABC-GHK = DEF × {(AG + BH + CK). SIMILAR POLYEDRONS DEFINITIONS 679 Similar polyedrons are polyedrons which have the same number of faces respectively similar and similarly placed, and which have their corresponding polyedral angles equal. 680 Homologous faces, lines, and angles of similar polyedrons. are faces, lines, and angles similarly situated. PROPOSITION XXIV. THEOREM 681 Two similar polyedrons may be divided into the same number of tetraedrons, similar each to each, and similarly placed. HYPOTHESIS. Let P and P' denote the similar polyedrons ABCDE-G and A'B'C'D'E'-G'. CONCLUSION. P and P'may be divided into the same number of tetraedrons, similar each to each, and similarly placed. PROOF Let B and B' be any two homologous triedral angles, and through the extremities of their edges, A, G, C, and A', G', C', respectively, pass planes. In the tetraedrons B-AGC and B'-A'G'C', the faces GAB, GBC, ABC, are similar to the faces G'A'B, G'B'C', A'B'C', respectively. AG AB GC § 349 .. the faces GAC and G'A'C' are similar, § 355 and the homologous triedral angles of these two tetraedrons are equal. § 592 .. the tetraedrons B-AGC and B'—A'G'C' are similar. § 679 If the tetraedrons B-AGC and B'-A'G'C' are removed, the remaining polyedrons are similar; for their corresponding new faces are similar, and their corresponding changed polyedral angles are equal since they have been equally reduced. Ax. 4 In like manner other corresponding similar tetraedrons may be removed until the polyedrons are divided into the same number of tetraedrons, similar each to each, and similarly placed. Q. E. D. 682 COROLLARY 1. The homologous edges of similar polyedrons are proportional. § 349 683 COROLLARY 2. Any two homologous lines in two similar polyedrons have the same ratio as any two homologous edges. § 349 684 COROLLARY 3. Any two homologous faces of two similar polyedrons have the same ratio as the squares of any two homologous edges. § 417 685 COROLLARY 4. The entire surfaces of two similar polyedrons have the same ratio as the squares of any two homologous edges. § 336 PROPOSITION XXV. THEOREM 686 The volumes of two similar tetraedrons are to each other as the cubes of their homologous edges. HYPOTHESIS. O-ABC and O'-A'B'C' are similar tetraedrons, V and V' their volumes. EXERCISE 1227. If in the above figure OA = 5, and O'A' = 3, find 1 The ratio of V to V'. 2 The ratio of the entire surfaces of V and V'. PROPOSITION XXVI. THEOREM 687 The volumes of two similar polyedrons are to each other as the cubes of any two homologous edges. HYPOTHESIS. P and P' are similar polyedrons, V and V' their volumes, GB and G'B' any two homologous edges. CONCLUSION. V:V' = GB3: G'B'3. PROOF Separate P and P' into tetraedrons, similar each to each, and similarly placed. § 681 Let T, T1, T2, ......., T', T'1, T'2, ......., denote the volumes of these similar tetraedrons respectively. |