This section is from the book "Spons' Mechanics' Own Book: A Manual For Handicraftsmen And Amateurs", by Edward Spon. Also available from Amazon: Spons' Mechanics' Own Book.
The solidity of a cone equals 1/3 the product of the area of the base multiplied by the perpendicular height; the convex surface equals half the product of the circumference of the base (diameter X 3.1416) multiplied by the slant height; the slant surface of a truncated (the top cut off) cone equals half the product of the sum of the circumferences of the 2 ends multiplied by the slant height.
To strike out a sheet to cover a whole cone, describe an are equal in length to the desired circumference, and at the radius of the required height. In Fig. 176, a is the desired cone, having a circumference at the base e of 15 in., and a height d e of 8 in.; then the length between b c must be 15 in., and the length between d e 8 in.
When only a frustrum of a cone is required, as for instance a funnel fitted over a pipe end, or the shoulder top of a can, the same law holds good; but in this case a second are must be described equal in length to the smaller circumference. Thus, in Fig. 177, supposing the ring a to have a larger circumference of 12 in. at the base, and a smaller circumference of 10 in. at the top, with a height of 7 in.; then 2 arcs have to he described at radii 7 in. apart, from the centre b (which is the point where the sides of a would cut each other if prolonged), the larger are c measuring 12 in. long, and the smaller d 10 in. Fig. 178 is another example where the shoulder has a much shallower slope, and when consequently the inner are d is much smaller than the outer c.