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82 can be called the arc of stretchout. The intersections on the arc of stretchout are numbered to correspond to the profile. Points 1 are connected with the apex and ¼-inch edges are added for a tin lock. Notice that the locks are parallel to lines number 1 and do not connect with the apex until they are "notched." The notch at the apex of any cone is made very long in order to bring the cone to a sharp point. The elements of the surface should be drawn, as shown in Fig. 117, and the length of these compared with the length of the corresponding foreshortened elements of Fig. 116.

38. Related Mathematics on Conical Flower Holder.—Area of a Sector.—A sector is a part of a circle set off by two radii and an arc. Is Fig. 117 a sector? What is the length of its radius? What is the length of its arc? If the length of the arc is multiplied by one-half of the radius, the result will be the area of a sector. Thus the area of a sector whose arc measures 14" and whose radius is 7" would equal 14×3½=49 sq. in.

The length of the arc, Fig. 117, is equal to the circumference of the base of the cone (the profile). In addition, the radius of pattern, Fig. 117, is equal to the slant height of the cone, Fig. 116. Because of these facts, the formulæ for the area of a sector and for the lateral area of a right cone are very much alike.

Lateral Area of a Cone.—The lateral area of a right cone is equal to the circumference of the base times one-half the slant height.

Suppose the base of a cone is 5" in diameter and the slant height is 12". To find the lateral area, the circumference, which would equal 5X3.1416=15.7080", must first be found.

Problem 19A.—How many square inches of surface area (lateral) has a right cone whose base is 7" in diameter and whose slant height is 11¼"?

Problem 19B.—The base of a cone has a circumference of 96" and a slant height of 102½". What is the area of its lateral surface in square inches?

Problem 19C.—How much would 1000 flower holders, Fig. 115, weigh if made from No. 28 galvanized steel (.7812 lb. per square foot)? Allow 5 per cent of total weight for waste.