601. The volume of any prism is equal to the product of its base by its altitude. E' E Given v denoting the volume, B the base, and H the altitude of any prism. Proof. Pass planes through one lateral edge BB', and the diagonals BE, BD, etc., of the base. The prism will be divided into triangular prisms, all of which will have the altitude H. 602. COR. 1. Prisms that have equivalent bases are to each other as their altitudes. 603. COR. 2. Prisms that have equal altitudes are to each other as their bases. 604. COR. 3. Prisms that have equal altitudes and equivalent bases are equivalent. Ex. 1527. Find the volume of a regular quadrangular prism if the side of its base is 4, and the lateral edge is 6. Ex. 1528. Find the volume of a regular hexagonal prism if the side of its base is 2, and the lateral edge is 5. Ex. 1529. Find the altitude of a regular hexagonal prism, if the side of the base equals 4, and the volume equals 60 √3. Ex. 1530. Find the volume of a regular triangular prism if the side of its base is 8, and the altitude is 10. Ex. 1531. Find the volume of a right triangular prism, if the lateral edge is 9, and the sides of the base are 6, 8, and 10. Ex 1532. E is the lateral edge and ABCD the base of a right quadrangular prism. Find the volume if AB = 9, BC = 12, CD = 14, DA = 13, AC 15, and E = 10. = Ex. 1533. Find the volume of a triangular prism, if the sides of its base are 6, 4, and 4, the lateral edge E = 8, and the inclination of E to the base is 30°. Ex. 1534. E is the lateral edge, p the projection of E upon the base, and s the side of the base of a prism. Find the volume, if E = 7, p = 1, 1, and the base is a regular hexagon. s = Ex. 1535. In the diagram for Prop. XIV, find AA', if V: ABCDE 25, and the projection of AA' upon the base is 6. == 200, Ex. 1536. In the diagram for Prop. XIV, find the volume of the prism, if ABCDE=15, the projection of AA' upon the base equals 3, and the inclination of AA' to the base equals 45°. Ex. 1537. In the same diagram find the inclination of AA' to the base, if AA' = 6, ABCDE = 20, and V = 60. Ex. 1538. The diagonals of a parallelopiped bisect each other. Ex. 1539. Find the number of cubic feet of air in a building whose dimensions are given in the annexed diagram. 33 18 50 30 Ex. 1540. What is the total surface of the house described in the preceding exercise? PYRAMIDS 605. DEF. A pyramid is a polyhedron bounded by the faces of a polyhedral angle and a plane. Obviously, all faces meeting at the vertex of the polyhedral angle are triangles, while the remaining surface is a polygon. 606. The vertex of the pyramid is the vertex of the polyhedral angle, the lateral edges are the edges of the polyhedral angle, the lateral faces are the triangles that meet at the vertex, the lateral area is the sum of the areas of the lateral faces, and the base is the face opposite the vertex. 607. DEF. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, a quadrilateral, etc. NOTE. A tetrahedron is a triangular pyramid. 608. DEF. The altitude of a pyramid is the length of the perpendicular from the vertex of the pyramid to the plane of the base, as VF. -- W 609. DEF. A regular pyramid is a pyramid whose base is a regular polygon, the center of which coincides with the foot of the altitude. A regular pyramid is also called a right pyramid. 610. DEF. The axis of a regular pyramid is its altitude. The lateral edges of a regular pyramid are equal since they cut off equal distances from the foot of the altitude. Hence the lateral faces of a regular pyramid are equal isosceles triangles. 611. DEF. The slant height of a regular pyramid is the altitude of any one of the lateral faces, as wE. (Diagram on p. 359.) 612. DEF. A truncated pyramid is the portion of a pyramid included between the base and a section formed by a plane cutting all the lateral edges. 613. DEF. A frustum of a pyramid is a truncated pyra mid in which the plane of the section is parallel to the base. 614. DEF. The altitude of a frustum is the perpendicular distance between its bases. 615. DEF. The lateral area of the frustum is the sum of the areas of its lateral faces; the lateral faces of a frustum of a regular pyramid are equal isosceles trapezoids. 616. DEF. The slant height of a frustum of a regular pyramid is the altitude of one of these trapezoids which form the lateral faces. PROPOSITION XV. THEOREM 617. The lateral area of a regular pyramid is equal to half the product of the slant height by the perimeter of the base. Given V-ABCDE, a regular pyramid of n lateral faces, P the perimeter of the base, L the lateral area, and s the slant height. 618. COR. The lateral area of a frustum of a regular pyramid is equal to one half the sum of the perimeters of the bases multiplied by the slant height. HINT. What is the shape of the lateral faces? 619. For many calculations relating to regu lar pyramids three types of right triangles are of importance. (See the figure for Prop. XV.) These types are illustrated by triangles VKF, VAK, and VFC. |