PRISMS 598 A prism is a polyedron two of whose faces are equal and parallel polygons, and whose other faces are parallelograms. 599 The bases of a prism are two equal parallel faces; the lateral faces are all the faces except the bases; the lateral edges are the intersections of the lateral faces; the base edges are the intersections of the bases with the lateral faces; and the lateral area is the sum of the areas of the lateral faces. 600 Prisms are triangular, quadrangular, etc., according as their bases are triangles, quadrilaterals, etc. 601 The altitude of a prism is the perpendicular between the planes of its bases. 602 A right prism is a prism whose lateral edges are perpendicular to its bases. An oblique prism is a prism whose lateral edges are oblique to its bases. 603 A regular prism is a right prism whose bases are regular polygons. 604 A truncated prism is a portion of a prism included between a base and a section formed by a plane oblique to the base and cutting all the lateral edges. 605 COROLLARY. The lateral edges of a prism are equal and parallel; the lateral edges of a right prism are equal to its altitude; and the lateral faces of a right prism are rectangles. 606 A parallelopiped is a prism whose bases are parallelograms. 607 An oblique parallelopiped is a parallelopiped whose lateral edges are oblique to its bases. 608 A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases. A rectangular parallelopiped is a right parallelopiped whose bases are rectangles. 610 A cube is a parallelopiped all of whose faces are squares. 611 A right section of a prism is a section formed by a plane perpendicular to its lateral edges. 612 The volume of a solid is the number of units of volume which it contains. In applied Geometry the unit of volume is usually a cube whose edge is some linear unit; as, a cubic inch, a cubic foot, etc. 613 Two solids having equal volumes are equivalent. PROPOSITION I. THEOREM 614 Sections of a prism made by parallel planes cutting all the lateral edges are equal polygons. M EL 'N HYPOTHESIS. ABCDE and A'B'C'D'E' are sections of the prism MN, formed by parallel planes cutting all the lateral edges. CONCLUSION. ABCDE and A'B'C'D'E' are equal polygons. PROOF AB is to A'B', BC is || to B'C', etc. = .. Z ABC · Z A'B'C', ≤ BCD = ≤ B'C'D', etc. $543 $549 $ 202 ... ABCDE and A'B'C'D'E' are mutually equilateral and equiangular and can be made to coincide. .. ABCDE = A'B'C'D'E'. $ 89 Q. E. D. 615 COROLLARY 1. Any section of a prism made by a plane parallel to the base is equal to the base. 616 COROLLARY 2. All right sections of a prism are equal. EXERCISES 1185 Any section of a prism made by a plane parallel to a lateral edge is a parallelogram. 1186 In the above figure, what kind of solid is MD' ? PROPOSITION II. THEOREM 617 The lateral area of a prism is equal to the product of a lateral edge by the perimeter of a right section. HYPOTHESIS. AD' is a prism, S its lateral area, E a lateral edge, and P the perimeter of the right section FK. CONCLUSION. SEX P. PROOF The face AB' is a parallelogram, whose base is AA' and altitude FG. .. area AB'=AA'x FG = Ex FG. Likewise area BC' BB' x GH = E× GH, etc. Adding, S= E(FG+GH + HK + etc.). 618 COROLLARY. = .. S Ex P. § 611 § 401 § 605 Q. E. D. The lateral area of a right prism is equal to the product of the altitude by the perimeter of the base. EXERCISES 1187 Find the lateral area of a prism, if the lateral edge is 18 inches and the perimeter of a right section is 32 inches. 1188 Find the lateral area of a right prism, if the altitude is 21 inches and the perimeter of the base is 38 inches. 619 Two prisms are equal, if the three faces including a triedral angle of the one are respectively equal to the three faces including a triedral angle of the other, and are similarly placed. HYPOTHESIS. In the prisms AJ and A'J', the faces AD, AG, and AK are respectively equal to the faces A'D', A'G', and A'K', and similarly placed. CONCLUSION. The prisms AJ and A'J' are equal. PROOF The triedral angles A and A' are equal, and the upper bases FJ and F'J' are equal. Apply the triedral angle A to the triedral angle A'. $592 $ 599 The faces AD, AG, and AK coincide respectively with the faces A'D', A'G', and A'K'; and the points K, F, and G coincide respectively with the points K', F', and G'. .. the upper bases FJ and F'J' coincide. .. the prisms coincide and are equal. § 515 Q. E. D. 620 COROLLARY 1. Two truncated prisms are equal, if the three faces including a triedral angle of the one are respectively equal to the three faces including a triedral angle of the other, and are similarly placed. 621 COROLLARY 2. Two right prisms are equal, if they have equal bases and equal altitudes. |