BOOK VII POLYHEDRONS, CYLINDERS, AND CONES POLYHEDRONS 557. DEF. A polyhedron is a solid bounded by planes. The faces of a polyhedron are the bounding planes; the edges are the intersections of the faces; and the vertices are the intersections of the edges. 558. DEF. A tetrahedron is a polyhedron of four faces; a hexahedron, one of six faces; an octahedron, one of eight faces; a dodecahedron, one of twelve faces; an icosahedron, one of twenty faces. NOTE. The least number of faces that a polyhedron can have is four; for three planes intersecting in a common point form a trihedral angle, and, therefore, one more plane is needed to form a solid. 559. DEF. A diagonal of a polyhedron is the straight line joining any two vertices not in the same face. 560. DEF. A convex polyhedron is one, every section of which is a convex polygon. NOTE. All the polyhedrons treated of in this book are convex. PRISMS AND PARALLELOPIPEDS 561. DEF. A prismatic surface is a surface generated by a moving straight line which continually intersects the boundary of a given polygon (as ABCDE) and is parallel to a fixed straight line not in the plane of the polygon. B C 562. DEF. A prism is a polyhedron bounded by a prismatic surface and two parallel planes. A The faces made by the parallel planes are the bases; the faces made by the prismatic surface are the lateral faces. The lateral edges are the intersections of two lateral faces. The lateral area is the sum of the areas of the lateral faces. The altitude of a prism is the perpendicular distance between the planes of the bases. Obviously all lateral faces are parallelograms (483) and the lateral edges are equal and parallel. 563. DEF. A right prism is a prism whose lateral edges are perpendicular to the planes of the bases. In a right prism all lateral faces are rectangles which are perpendicular to the base, and all lateral edges are equal to the altitude. 564. DEF. A regular prism is a right prism whose bases are regular polygons. 565. DEF. An oblique prism is a prism whose lateral edges are not perpendicular to the planes of the bases. A prism is triangular, quadrangular, etc., according as its bases are triangles, quadrilaterals, etc. 566. DEF. A truncated prism is the part of a prism included between a base and a section made by a plane not parallel to the base and cutting all of the lateral edges. 567. DEF. A right section of a prism is a section made by a plane perpendicular to all the lateral edges of the prism (prolonged if necessary). If a plane is perpendicular to one lateral edge of a prism, it is perpendicular to every lateral edge (504). 568. DEF. A parallelopiped is a prism whose bases are parallelograms; hence one, all of whose faces are parallelograms. 569. DEF. A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to the bases. Obviously a parallelopiped is a right parallelopiped if one of its lateral edges is perpendicular to the base. 570. DEF. A rectangular parallelopiped is a right parallelopiped whose bases are rectangles; hence all of whose faces are rectangles. To prove that a parallelopiped is rectangular it is only necessary to show that the three angles at any one vertex are right angles. 571. DEF. A cube is a parallelopiped all of whose faces are squares. Ex. 1495. How many diagonals can be drawn in (a) a parallelopiped? (b) a pentagonal prism? (c) quadrangular pyramid ? Ex. 1496. What is the least number of faces that a polyhedron can have? The least number of edges? of vertices? PROPOSITION I. THEOREM 572. The sections of a prism made by parallel planes cutting all the lateral edges are congruent polygons. Given KM, a prism intersected by parallel planes AD and A'D', cutting all the lateral edges. ∠ABC = ∠ A'B'C', ∠ BCD = ∠ B'C'D', etc. (495) (144) AB = A'B', BC = B'C', CD = C'D', etc. 573. COR. The bases of a prism are congruent, and every section of a prism made by a plane parallel to the base is congruent to the base. PROPOSITION II. THEOREM 574. The lateral area of a prism is equal to the product of the perimeter of a right section and a lateral edge. Given AD' a prism, L its lateral area, E a lateral edge, and P the perimeter of right section GK. 575. COR. The lateral area of a right prism equals the perimeter of the base multiplied by the altitude. Ex. 1497. Find the lateral area of a right prism if its altitude is 15 in., and its base is a triangle the sides of which are 8 in., 10 in., and 11 in. respectively. Ex. 1498. Find the altitude of a right prism if its lateral area equals 190 sq. in., and its base is a quadrilateral the sides of which are 7 in., 8 in., 11 in., and 12 in. respectively. Ex. 1499. Find perimeter of a right section of a prism, if the lateral area equals 40, the projection of a lateral edge upon the base equals 4, and the altitude of the prism equals 3. (See 598.) Ex. 1500. Find the altitude of a right prism if its base is an equilateral triangle inscribed in a circle of radius 5 in., and its lateral area is 135 sq. in. |