Place the parallelopipeds so that the edge CF may be common, and the right dihedral angles at CF vertical. Produce the faces AG, BG, AF, DI, so as to meet, and form a third rectangular parallelopiped, CH. Since CA, CH, have the same base, FG, and the altitudes BC, CI, CA: CH BC: CI = BC CG: CI· CG. (526, 318) Since CD, CH, have the same base, FI, and altitudes CK, CG, CD:CH= =CK: CG = CI · CK: CI · CG; (526, 318) .. CA: CD = rect. BC CG rect. CI CK; : (249) i.e., p'ped CA: p'ped CD = base BG: base CE. Q.E.D. 529. SCHOLIUM. The foregoing proposition may be expressed as follows: If two rectangular parallelopipeds have one dimension in common, they are to each other as the products of their other two dimensions. PROPOSITION IX. THEOREM. 530. Rectangular parallelopipeds are to each other as the products of their three dimensions. Given: Two rectangular parallelopipeds, P, P', with dimensions a, b, c, and a', b', c', respectively; To Prove: P: Paxbxc: a' x b'x c'. Let be a third rectangular parallelopiped whose dimensions are a', b, c. Since P, Q, have two dimensions b, c, in common, (Hyp.) P: Q=a: a'. (527) Since Q, P', have the dimension a' in common, (Hyp.) Q: P'bx c : b' x c'. ... P: P' = a xbx c: a' x b' x c'. Q.E.D. (529) (242) 531. DEFINITION. A cube is a rectangular parallelopiped whose faces are all squares. COR. The edges of a cube are all equal. 532. DEFINITION. The unit of volume is the cube whose edge is the linear unit, and whose base is, consequently, the unit of area. 533. COR. 1. The volume of a rectangular parallelopiped is measured by the product of its three dimensions. For if a, b, c, be the dimensions of a rectangular parallelopiped P, and U be the unit cube, then (530), This could be expressed at greater length as follows: The number of unit cubes in any rectangular parallelopiped is equal to the number of units in the product of the numerical measures of its length, breadth, and thickness. 534. COR. 2. The volume of a cube is measured by the cube of its edge. Thus if the edge of a cube be 7 linear units, the cube contains 73 = 343 unit cubes; if the edge be a linear units, the cube contains a3 times the unit cube. Geom. - 18 535. The volume of any prism is measured by the product of its base and altitude. 1o. Any parallelopiped is equivalent to a rectangular parallelopiped having the same altitude and an equivalent base (525); and the volume of the latter is measured by the product of its three dimensions, that is, of its base and altitude (533); hence the volume of any parallelopiped is measured by the product of its base and altitude. 2o. Any triangular prism is equivalent to one half the parallelopiped having the same altitude and a base of twice the area (524); now, the volume of the latter being measured by the product of its base and altitude (1°), the volume of the triangular prism is also measured by the product of its base and altitude. 3°. By passing planes through its lateral edges, any prism can be divided into triangular prisms whose altitudes are the same as that of the given prism, and whose triangular bases together form the base of the given prism. As the volume of each of these triangular prisms is meas ured by the product of its base and altitude (2°), the volume of any prism is measured by the product of its base and altitude. Q.E.D. 536. COR. Prisms having equivalent bases are to each other as their altitudes; prisms having equal altitudes are to each other as their bases; and prisms are to each other as the products of their bases and altitudes. EXERCISE 732. Two triangular prisms, A and B, have the same altitude. A has for base a right-isosceles triangle; B, for base an equilateral triangle of side equal to the hypotenuse of the base of A. Find the ratio of the volume of A to that of B. 733. Find the ratio of the lateral area of A to that of B. PYRAMIDS. 537. A pyramid is a polyhedron bounded by a polygon called the base, and by triangular planes meeting in a common point called the vertex. A plane intersecting the faces of any polyhedral angle cuts off a pyramid. The terms lateral face, lateral surface, lateral edge, basal edge, are defined as for prisms (504). 538. The altitude of a pyramid is the perpendicular distance from its vertex to the base; as SP. 539. A regular pyramid has for base a regular polygon, and has its vertex in the perpendicular at the center of the base, which perpendicular is called the axis of the pyramid. E S P: A B 540. The slant height of a regular pyramid is the altitude of any lateral face. 541. A pyramid is triangular, quadrangular, pentagonal, etc., according as its base is a triangle, quadrangle, pentagon, etc. In the triangular pyramid, or tetrahedron (499), any one of the faces may be regarded as the base. 542. A truncated pyramid is the portion of a pyramid included between the base and a plane that intersects all the lateral faces. 543. A frustum of a pyramid is a trun cated pyramid in which the intersecting plane is parallel to the base. The base of the pyramid is called the lower base of the frustum; the parallel section, the upper base. 544. The altitude of a frustum is the perpendicular distance between its bases; the slant height of a frustum of a regular pyramid is the altitude of any lateral face. PROPOSITION XI. THEOREM. 545. If a pyramid be cut by a plane parallel to the base: 1o. The edges and altitude will be divided proportionally. 2°. The section is a polygon similar to the base. Given: A pyramid S-ABD, whose altitude SP is cut in p by a plane abd parallel to the base; To Prove: 1°, SA: Sa = SB: sb = SP: Sp, etc.; 2°, abd is similar to ABD. 1o. Suppose a plane passed through S || to ABD. Since the edges and altitude are cut by || planes, (Hyp. and Const.) SA: Sa = SB: sb = SC: Sc = SP: Sp, etc. Q.E.D. (459) Since plane abd is to plane ABD, 2o. (Hyp.) ab is to AB, bc is | to BC, cd is | to CD, etc., (451) (115) (455) Since ab is to AB, and be is to BC, ▲ sab is sim. to ▲ SAB, and ▲ sbc to ▲ SBC; (291) .. ab: AB = sb: SB, and bc: BC = sb: SB; (284) |