PROPOSITION XXV. THEOREM. 663. The sum of the face angles of any polyhedron is equal to four right angles taken as many times, less two, as the polyhedron has vertices. P Let E denote the number of edges, V the number of vertices, F the number of faces, and S the sum of the face angles, of the polyhedron P. Proof. Since E denotes the number of edges, 2 E will denote the number of sides of the faces, considered as independent polygons, for each edge is common to two polygons. If an exterior angle is formed at each vertex of every polygon, the sum of the interior and exterior angles at each vertex is 2 rt. (§ 86); and since there are 2 E vertices, the sum of the interior and exterior angles of all the faces is 2 Ex 2 rt. 4, or Ex 4 rt. s. But the sum of the ext. s of each face is 4 rt. s. FX 4 rt. s. SEX 4rt. - FX 4 rt. s =(EF) 4 rt. 4. § 207 Therefore, But that is, E-FV - 2. Therefore, S=(V2) 4 rt. . Q.E. D. SIMILAR POLYHEDRONS. 664. DEF. Similar polyhedrons are polyhedrons that have the same number of faces, respectively similar and similarly placed, and have their corresponding polyhedral angles equal. 665. DEF. Homologous faces, lines, and angles of similar polyhedrons are faces, lines, and angles similarly situated. PROPOSITION XXVI. THEOREM. 666. Two similar polyhedrons may be decomposed into the same number of tetrahedrons similar, each to each, and similarly placed. Let P and P' be two similar polyhedrons with G and G' homologous vertices. To prove that P and P' can be decomposed into the same number of tetrahedrons, similar each to each, and similarly placed. Proof. Divide all the faces of P and P', except those which include the angles G and G', into corresponding A. Pass planes through G and the vertices of the A in P; also through G' and the vertices of the A in P'. Any two corresponding tetrahedrons G-ABC and G'-A'B'C' are similar; for they have the faces ABC, GAB, GBC, similar, respectively, to A'B'C', G'A'B', G'B'C' ; and the face GAC similar to G'A'C', § 365 $ 358 equal. § 582 They also have the corresponding trihedral .. the tetrahedron G-ABC is similar to G'-A'B'C'. § 664 If G-ABC and G'-A'B'C' are removed, the polyhedrons remaining continue similar; for the new faces GAC and G'A'C' have just been proved similar, and the modified faces AGF and A'G'F', CGH and C'G'H', are similar (§ 365); also the modified polyhedral G and G', A and A', C and C', remain equal each to each, since the corresponding parts taken from them are equal. The process of removing similar tetrahedrons can be carried on until the polyhedrons are decomposed into the same number of tetrahedrons similar each to each, and similarly placed. Q. E. D. 667. COR. 1. The homologous edges of similar polyhedrons are proportional. $ 351 668. COR. 2. Any two homologous lines in two similar polyhedrons have the same ratio as any two homologous edges. $ 351 669. COR. 3. Two homologous faces of similar polyhedrons are proportional to the squares of two homologous edges. § 412 670. COR. 4. The entire surfaces of two similar polyhedrons are proportional to the squares of two homologous edges. § 335 671. The volumes of two similar tetrahedrons are to each other as the cubes of their homologous edges. Let V and V' denote the volumes of the two similar tetrahedrons S-ABC and S'-A'B'C'. Ex. 676. The homologous edges of two similar tetrahedrons are as 6: 7. Find the ratio of their surfaces and of their volumes. Ex. 677. If the edge of a tetrahedron is a, find the homologous edge of a similar tetrahedron twice as large. PROPOSITION XXVIII. THEOREM. 672. The volumes of two similar polyhedrons are to each other as the cubes of any two homologous edges. Let V, V' denote the volumes, GB, G'B' any two homologous edges, of the polyhedrons P and P'. Proof. Decompose these polyhedrons into tetrahedrons similar, each to each, and similarly placed. Denote the volumes of these tetrahedrons by v, v1, v2, § 666 |