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 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 A 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. 2o. 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 2°. planes, (Hyp. and Const.) SA: Sa = SB: Sb = SC: Sc = SP: Sp, etc. Q.E.D. (459) Since plane abd is | to plane ABD, (Hyp.) ab is to AB, bc is to BC, cd is to CD, etc., (451) and they are similarly directed, abd and ABD are mutually equiangular. (115) (455) (291) sb: SB; (284) Since ab is to AB, and bc is to BC, .. ab: AB = sb:SB, and bc: BC: 546. COR. 1. The area of any section of a pyramid parallel to the base, is proportional to the square of its distance from the vertex. For parallel sections being similar to the base (545), their areas are proportional to the squares of their homologous sides (344). Thus -2 -2 -2 abd: ABD = ab2: AB = Sa: SA Sp: SP. 547. COR. 2. If two pyramids, S-ABD, S'-A'B'D', having equal altitudes, SP, S'P', are cut by planes parallel to their bases, and at equal distances, Sp, s'p', from their vertices, the sections, abd, a'b'd', will be to each other as the bases. 548. COR. 3. If two pyramids have equal altitudes and equivalent bases, sections made by planes parallel to their bases, and at equal distances from the vertices, are equivalent. EXERCISE 734. Show that a plane perpendicular to the axis of a regular pyramid forms equal dihedral angles with all the faces of the pyramid. 735. In order that a plane intersecting the faces of a polyhedral angle may cut off a regular pyramid, what conditions must be fulfilled 、 in regard to the form of the polyhedral and the inclination of the plane ? 736. In the diagram for Prop. XI., if a plane be passed through the mid point of pP, parallel to the base, show that the perimeter of the section thus formed will be equal to half the sum of the perimeters of ABD and abd. PROPOSITION XII. THEOREM. 549. The lateral surface of a regular pyramid is equivalent to one half the rectangle contained by its perimeter and slant height. Given: A regular pyramid S-ABD, and SF its slant height; To Prove: The lateral surface of S-ABD is equivalent to since A, B, C, D, E, are equally distant from P, (539) .. isos. A SAB=isos. A SBC=isos. A SCD, etc. But A SAErect. SF · AE; i.e., lat. surf. of S-ABD SF perimeter. (437) (69) (331) (335) Q.E.D. 550. COR. 1. The lateral surface of a frustum of a regular pyramid is equivalent to one half the rectangle contained by the slant height of the frustum and the sum of the perimeters of the bases. (338) For it is the sum of as many trapezoids as the base has sides, having for common altitude the slant height of the frustum. (544) 551. COR. 2. The dihedral and trihedral angles at the base of a regular pyramid are all equal. |