19. What is the total surface of the tank of a sprinkling wagon which is 3 ft. in diameter and 7 ft. long? 20. How much paint will be required for the lateral surface of a cylindrical silo which is 16 ft. in diameter and 30 ft. high, allowing 1 gallon to 250 sq. ft.? 21. What is the grinding surface of a grindstone that is 2 ft. in diameter and 3 in. thick? 22. A room is heated by 420 ft. of pipe whose outer diameter is 3 in. What is the total heating surface? 47. Volume and surface of a pyramid. A pyramid is a solid whose base is any polygon and whose lateral surfaces are triangles with a common vertex, which is called the vertex of the pyramid. The altitude of a pyramid is the perpendicular from the vertex to the base. If the base of a pyramid is a regular polygon and if the edges meeting at the vertex are all equal, the pyramid is called a regular pyramid. The perpendicular from the vertex to one edge of the base of a regular pyramid is called the slant height of the pyramid. In Figure 58, O is the vertex, OP the altitude, and OM the slant height of the regular pyramid. Take a hollow prism and a hollow pyramid which have equal bases and equal altitudes. It may be found by trial that the prism will hold three times as much sand or other substance as the pyramid. Since the volume of the prism is the product of its base and altitude, we have this Pyramid FIG. 58 Rule. The volume of a pyramid is one third of the product of its base and altitude. If V is the volume, h the altitude, and b the area of the base, we have the formula bh V= 3 Cut a form like Figure 59 from light cardboard or stiff paper. Fold along the dotted lines, paste together the edges, and thus form a pyramid. What kind of pyramid does this give? The lateral faces of a regular pyramid are what kind of triangles? Are all the faces of the same size? In Figure 59 what is the area of one lateral face? What is the total lateral area? FIG. 59 What is the perimeter of the base? What must the perimeter of the base be multiplied by to give the lateral area? If A is the lateral area of a regular pyramid, s the slant height, and p the perimeter of the base, then A = sp = sp. 2 Exercise 50 1. Find the volume of a pyramid whose base is 185 sq. in. and whose altitude is 18 in. 2. Find the volume of a pyramid whose base is a square with an edge of 15 in. and whose altitude is 8 in. 3. Is the pyramid in the last exercise necessarily a regular pyramid? UNIV. OF AMID: · 101· VOLUME AND SURFACE OF A PYRAMID 4. Do you know of any examples of pyramids in the roofs of houses or in the tops of towers? 5. A brass paper weight has the form of a pyramid. Its base is 1 inch square and its altitude is 18 in. How many pounds of brass are required to make 100 such paper weights? The specific gravity of brass is 8.4. 6. The roof of a building 24 ft. square is in the form of the surface of a regular pyramid, whose slant height is 14.5 ft. The roof is covered with a composition roofing. How many square yards of this roofing are required if there is a waste of 5% of the 8. The pyramid of Cheops in Egypt has a base 764 ft. square and an altitude of 480.75 ft. Find its volume. 9. How many square yards of canvas are required to make a tent in the form of a pyramid whose base is 10 ft. square and whose height is 9 ft.? -12' FIG. 61 10. How many square yards of canvas are required to make a square tent of the dimensions given in Figure 61? 11. Solve the formula V=3bh for h. 12. Find the altitude of a pyramid whose volume is 84 cu. in. and whose base is 6 in. square. 13. The base of a pyramid is an equilateral triangle whose side is 10 in. The volume of the pyramid is 5196 cu. in. Find the altitude. 14. Solve the formula V=3bh for b. 15. Find the area of the base of a pyramid whose altitude is 14 ft. and volume 264 cu. ft. 48. Volume and surface of a cone. If a right triangle is rotated about the altitude as an axis, the figure traced is called a cone. The kind of cone thus obtained is a right circular cone. It is the only kind of cone that we shall consider in this book. When we speak of a cone, we shall mean a right circular cone. Cone The point A is the vertex of the cone. The circle with the center O is called the base of the cone, and the other surface of the cone is called the lateral surface. The perpendicular distance from the vertex to the base is the altitude of the cone. The distance from the vertex to the edge of the base is the slant height of the cone. Take a hollow cylinder and a hollow cone with equal bases and equal altitudes. It may be found by trial that the cylinder will hold three times as much sand or other substance as the cone. If the cylinder has the base b and the altitude a, what is its volume? What is the volume of the cone with the same base and the same altitude? Make a rule for finding the volume of a cone when the area of the base and the altitude of the cone are known. If the radius of the base is r, what is the area of the base? The formula for the volume of a cone is V = r2h, where r is the radius of the base and h the altitude of the cone. Cut a form like Figure 63 from stiff paper or light cardboard. If the edges OA and OB are then pasted together may be found by the same method as was used for finding the area of a circle. Cut the sector AOB (Figure 63) into smaller sectors and fit them together as in Figure 64. Let the pupil show from this illustration that the area of a sector equals the radius times one-half the arc. FIG. 64 In Figure 63 the arc AB is seen to be the same length as the circumference of the base of the cone, and the radius OA is the slant height of the cone. We have then the Rule. The lateral surface of a cone equals one-half the product of the circumference of the base by the slant height. If the radius of the base of a cone is r, the slant height s, and the lateral surface L, then L=πrs. If S is the total surface of the cone, then |