Proposition 9. Theorem. 603. Two rectangular parallelopipeds having equal altitudes are to each other as their bases. Hyp. Let the rectangles ab and a'b' be the bases of two rectangular parallelopipeds, P and Q, having the common altitude c. P a a' R And because R and Q have the two dimensions b' and c in common, (Cons.) (602) Q.E.D. 604. SCH. This theorem may also be expressed as follows: Two rectangular parallelopipeds having one dimension in common are to each other as the products of the other two dimensions. EXERCISE. Find the length of the diagonal of a rectangular parallelopiped whose edges are 1, 4, and 8. Proposition 10. Theorem. 605. Any two rectangular parallelopipeds are to each other as the products of their three dimensions. Hyp. Let P and Q be two rectangular parallelopipeds whose dimensions are a, b, c, and a', b',c', respectively. Р And because R and Q have the two dimensions a', b' in common, Find the ratio of two rectangular parallelopipeds whose dimensions are 4, 7, 9, and 6, 14, 15, respectively. Proposition 11. Theorem. 606. The volume of a rectangular parallelopiped is equal to the product of its three dimensions. Hyp. Let P be the rectangular parallelopiped, a, b, and c its dimensions, and let Q be the cube whose edge is the linear unit. Q then is the unit of volume. P To prove Proof. (584) 607. SCH. The statement of this theorem is an abbreviation of the following: The number of units of volume in a rectangular parallelopiped is equal to the product of the numbers which measure the linear units in its three dimensions. Compare (361). When the three dimensions of a rectangular parallelopiped are each exactly divisible by the linear unit, the truth of the theorem may be shown by dividing the solid into cubes, each equal to the unit of volume. Thus, if AB contain the linear unit 3 times, AC, 4 times, and AD, 5 times, these edges may be divided respectively into 3, 4, and 5 equal parts, and then planes passed through the several points of division at right angles to these B edges will divide the solid into cubes each equal to the unit of volume. Hence the whole solid contains 3 × 4 × 5, or 60 cubes, each equal to the unit of volume. 608. COR. 1. Since a x b is the area of the base, and c is the altitude, of the parallelopiped P; therefore the above result may be expressed in the form: The volume of a rectangular parallelopiped is equal to the product of its base and altitude. 609. COR. 2. The volume of a cube is the third power of its edge, being the product of three equal factors; if the edge is 1, the volume is 1 x 1 X 1=1; if the edge is a, the volume is a X a × a = a3. Hence it is that in arithmetic and algebra the "cube" of a number is the name given to the "third power" of a number. EXERCISES. 1. Find the surface, and also the volume, of a rectangular parallelopiped whose edges are 4, 7, and 9 feet. 2. Find the surface of a rectangular parallelopiped whose base is 8 by 12 feet and whose volume is 384 cubic feet. 3. Find the volume of a rectangular parallelopiped whose surface is 208 and whose base is 4 by 6. 4. Find the length of the diagonal of a rectangular parallelopiped whose edges are 3, 4, and 5. 5. Find the ratio of two rectangular parallelopipeds whose dimensions are 1, 4, 8, and 3, 4, 5, respectively. 6. Find the surface of a rectangular parallelopiped whose base is 7 by 9 feet and whose volume is 315 cubic feet. 7. Find the volume of a rectangular parallelopiped whose surface is 416 and whose base is 4 by 12. 8. A man wishes to make a cubical cistern whose contents are 186624 cubic inches: how many feet of inch boards will line it? 9. Find the side of a cube which contains as much as a rectangular parallelopiped 20 feet long, 10 feet wide, and 6 feet high. Proposition 12. Theorem. 610. The volume of any parallelopiped is equal to the product of its base and altitude. The rt. parallelopiped EG', formed by the sections EE'H'H, FF'G'GL to the produced edge EF is equivalent to AC'. (590) Now produce the edges HE, GF, G'F', H'E', and take KL HE. The parallelopiped KNN'K'-L formed by the sections LMM'L', KNN'K' to the produced edge KL is equivalent to EG', and ... to AC'. (590) ... given parallelopiped AC' and the last, KM', are equivalent. But, since the rt. sections LM', KN', are rectangles, ... KM' is a rectangular parallelopiped. Also, since (582) area ABCD = area EFGH = area KLMN, (363) (360), and the three solids have the same altitude B'O, ... the volume KM' = KLMN × B'O. (523) (608) Q.E.D. |