PROPOSITION XIII. THEOREM. 628. The volume of any prism is equal to the product of its base by its altitude. Let V denote the volume, B the base, and H the altitude of the prism DA'. Proof. Planes passed through the lateral edge AA', and the diagonals AC, AD of the base, divide the given prism into triangular prisms that have the common altitude H. The volume of each triangular prism is equal to the product of its base by its altitude (§ 627); and hence the sum of the volumes of the triangular prisms is equal to the sum of their bases multiplied by their common altitude. But the sum of the triangular prisms is equal to the given prism, and the sum of their bases is equal to its base. Ax. 9 Therefore, the volume of the given prism is equal to the product of its base by its altitude. V = B× H. Q. E. D. That is, 629. COR. Two prisms are to each other as the products of their bases by their altitudes; prisms having equivalent bases are to each other as their altitudes; prisms having equal altitudes are to each other as their bases; prisms having equivalent bases and equal altitudes are equivalent. PROBLEMS OF COMPUTATION. ་ Ex. 643. If the edge of a cube is 15 inches, find the area of the total surface of the cube. Ex. 644. If the length of a rectangular parallelopiped is 10 inches, its width 8 inches, and its height 6 inches, find the area of its total surface. Ex. 645. Find the volume of a right triangular prism, if its height is 14 inches, and the sides of the base are 6, 5, and 5 inches. Ex. 646. The base of a right prism is a rhombus, one side of which is 10 inches, and the shorter diagonal is 12 inches. The height of the prism is 15 inches. Find the entire surface and the volume. Ex. 647. Find the volume of a regular prism whose height is 10 feet, if each side of its triangular base is 10 inches. Ex. 648. How many square feet of lead will be required to line a cistern, open at the top, which is 4 feet 6 inches long, 2 feet 8 inches wide, and contains 42 cubic feet? Ex. 649. An open cistern 6 feet long and 4 feet wide holds 108 cubic feet of water. How many square feet of lead will it take to line the sides and bottom? Ex. 650. An open cistern is made of iron 2 inches thick. The inner dimensions are: length, 4 feet 6 inches; breadth, 3 feet; depth, 2 feet 6 inches. What will the cistern weigh (i) when empty ? (ii) when full of water? (The specific gravity of the iron is 7.2.) Ex. 651. Find the volume of a regular hexagonal prism, if its height is 10 feet, and each side of the hexagon is 10 inches. Ex. 652. Find the length of an edge of a cubical vessel that will hold 2 tons of water. Ex. 653. One edge of a cube is a. Find the surface, the volume, and the length of a diagonal of the cube. Ex. 654. A diagonal of one of the faces of a cube is a. Find the volume of the cube. C. Ex. 655. The three dimensions of a rectangular parallelopiped are a, b, Find the area of its surface, its volume, and the length of a diagonal. Ex. 656. The volume of a parallelopiped is V, and the three dimensions are as m:n: p. Find the dimensions. PYRAMIDS. 630. DEF. A pyramid is a polyhedron of which one face, called the base, is a polygon of any number of sides and the other faces are triangles having a common vertex. The faces which have a common vertex are called the lateral faces of the pyramid, and their common vertex is called the vertex of the pyramid. Pyramids. 631. DEF. The intersections of the lateral faces are called the lateral edges of the pyramid. 632. DEF. The sum of the areas of the lateral faces is called the lateral area of the pyramid. 633. DEF. The altitude of a pyramid is the length of the perpendicular let fall from the vertex to the plane of the base. 634. DEF. A pyramid is called triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc. 637. DEF. The perpendicular let fall from the vertex to the base of a regular pyramid is called the axis of the pyramid. The lateral edges of a regular pyramid are equal, for they cut off equal distances from the foot of the perpendicular let fall from the vertex to the base. § 514 S eral faces of a regular pyramid are equal. S 638. DEF. The slant height of a regular pyramid is the altitude of any one of the lateral faces. It bisects the base of the lateral face in which it is drawn. § 149 639. DEF. A truncated pyramid is the portion of a pyramid included between the base and a section made by a plane cutting all the lateral edges. A frustum of a pyramid is the portion of a pyramid included between the base and a section parallel to the base. 641. DEF. The lateral faces of a frustum of a regular pyramid are equal isosceles trapezoids; and the sum of their areas is called the lateral area of the frustum. 642. DEF. The slant height of the frustum of a regular pyramid is the altitude of one of these trapezoids. PROPOSITION XIV. THEOREM. 643. The lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base. Let S denote the lateral area of the regular pyramid V-ABCDE, L its slant height, and P the perimeter of its base. Proof. The AVAB, VBC, etc., are equal isosceles A. § 637 The area of each ▲ is multiplied by its base. § 403 .. the sum of the areas of these A is L × P. But the sum of the areas of these A is equal to S, the lateral area of the pyramid. 644. COR. The lateral area of the frustum of a regular pyramid is equal to half the sum of the perimeters of the bases multiplied by the slant height of the frustum. § 407 Ex. 657. Find the lateral area of a regular pyramid if the slant height is 16 feet, and the base is a regular hexagon with side 12 feet. Ex. 658. Find the lateral area of a regular pyramid if the slant height is 8 feet, and the base is a regular pentagon with side 5 feet. Ex. 659. Find the total surface of a regular pyramid if the slant height is 6 feet, and the base is a square with side 4 feet. |