98 233, 234 POLYHEDRONS 171 II. POLYHEDRONS Proposition 1. The Polyhedron Theorem 234. Theorem. In a polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces. Given AG, a polyhedron; e, the number of edges; v, the number of vertices; and f, the number of faces. Prove that e+2=v+f. Proof. For one face, as BCGF, e=v. Adding a second face, as ABCD, there is formed a surface of two faces which has one edge (BC), and two vertices (B and C), common to the two faces. Hence for two faces, e=v+1. Adding a third face ABFE, adjoining each of the first two, this face will have two edges (AB, BF) and three vertices (A, B, F) in common with the surface of two faces. Hence for three faces, e=v+2. Now the addition of the next face, which is the last one, will not increase the number of edges or vertices. Hence for f faces, e=v+f-2, or e +2=v+f. Proposition 2. Truncated Triangular Prism 235. Theorem. A truncated triangular prism is equivalent to the sum of three pyramids whose common base is the base of the prism and whose vertices are the three vertices of the inclined section. Given the truncated triangular prism with base ABC and inclined section DEF, and divided into the three pyramids D-ABC, E-ABC, and F-ABC. Prove that ABC-DEF is equivalent to the sum of the three pyramids D-ABC, E-ABC, and F-ABC. Proof. Dividing the truncated prism ABC-DEF into the pyramids E-ACD, E-ABC, E-CFD, as shown in the second group of figures, we shall now show that these pyramids are equivalent to those in the first group. $$ 235-237 TRUNCATED PRISM 173 Now pyramid E-ACD= pyramid B-ACD, $ 124 because they have the common base ACD and equal altitudes, since the vertices E and B lie on EB which is 1l to the plane ACD. But the pyramid B-ACD may be regarded as having the base ABC and the vertex D; that is, as pyramid D-ABC. .. pyramid E-ACD= pyramid D-ABC. The pyramids E-ABC are the same in each division of the prism; that is, they have the base ABC and the vertex E. Now ACFD=ACFA, $ 26, 1 because they have the common base CF and equal altitudes, since their vertices lie on AD which is II to CF. Then pyramid E-CFD=pyramid B-CFA, $ 124 because they have equivalent bases (the ACFD and CFA) and equal altitudes, since EB is Il to the plane ACFD. But the pyramid B-CFA may be regarded as having the base ABC and the vertex F; that is, as pyramid F-ABC. .:. pyramid E-CFD= pyramid F-ABC. Hence the truncated triangular prism ABC-DEF is equivalent to the sum of the three pyramids whose common base is ABC and whose vertices are D, E, and F. 236. Corollary. The volume of a truncated right triangular prism is one third the product of the base and the sum of the lateral edges. D Since the lateral edges DA, EB, FC are 1 to the base ABC, they are the altitudes of the three pyramids whose sum is equivalent to ABC-DEF. A 'C It is interesting to consider the special case in which ADEF is II to AABC. B 237. Similar Polyhedrons. Polyhedrons which have the same number of faces, respectively similar and similarly placed, and which have their corresponding polyhedral angles equal, are called similar polyhedrons. 2 Proposition 3. Similar Polyhedrons 238. Theorem. Two similar polyhedrons can be sepato each and similarly placed. Given the similar polyhedrons P and P'. Prove that P and P' can be separated into the same number of tetrahedrons, similar each to each and similarly placed. Proof. Let E and E' be corresponding vertices. By drawing corresponding diagonals, as AC, A'C', let all the faces of P and P', except those which include the & E and E', be divided into corresponding A. Also, let a plane, as EAC, pass through E and each diagonal of the faces of P, and a plane, as E'A'C', through E' and each corresponding diagonal of P'. Any two corresponding tetrahedrons E-ABC and E'-A'B'C' have the faces ABC, EAB, EBC similar respectively to the faces A'B'C', E'A'B', E'B'C'. $ 24,8 AE AB AC BC Since EC $ 24, 1 A'B' A'C' B'C' E'C '' then face EAC is similar to face E'A'C'. $ 24,5 $8 238-243 SIMILAR POLYHEDRONS 175 Then the corresponding trihedral As of the tetrahedrons are equal. $ 84 Hence E-ABC is similar to E'-A'B'C'. $ 237 If E-ABC and E'-A'B'C' are removed, the polyhedrons which are left remain similar; for the new faces EAC and E'A'C' have just been proved similar, the modified faces AED and A'E'D', ECF and E'C'F' are similar ($24, 8), and the modified polyhedral As E and E', A and A', C and c' remain equal each to each, since the corresponding parts taken from these s are equal. This process of removing similar tetrahedrons can be continued as necessary. Hence P and P' can be separated into the same number of tetrahedrons, similar each to each and similarly placed. 239. Corollary. The corresponding edges of similar polyhedrons are proportional. This follows from the definitions of 88 237 and 24, 1. 240. Corollary. Any two corresponding lines in two similar polyhedrons have the same ratio as any two corresponding edges. For these lines may be shown to be sides of similar polygons. 241. Corollary. Two corresponding faces of similar polyhedrons are proportional to the squares of any two corresponding edges. This follows from $8 237 and 26, 3. 242. Corollary. The areas of the entire surfaces of two similar polyhedrons are proportional to the squares of any two corresponding edges. 243. Corollary. The areas of two similar cylinders, or of two similar cones, are proportional to the squares of any two corresponding lines. Consider limits, and apply § 242. |