mid S-ADF. If we denote by s the lateral area of the pyramid, by p the perimeter of its base, and by its slant height, whatever be the number of lateral faces of the pyramid. Conceive the number of lateral faces to be indefinitely increased by continually doubling the number of sides of the base polygon; then, since s, p, and I have for limits S, C, L, respectively, 685. COR. 1. If R denote the radius of the base, then C=2π· R (396), and s R • L) = π • R. L. = = 1 (2· π • Also, since the area of the base = · R2 (398), the total area, T, of the surface of a cone, is expressed by 687. COR. 2. The lateral areas, or the total areas, of similar cones of revolution, are to each other as the squares of their altitudes, or as the squares of their radii. This may be proved as was Cor. 3 of Prop. I. 688. DEFINITION. A truncated cone is the portion of a cone intercepted between the base of the cone and a plane cutting its lateral surface. 689. DEFINITION. A frustum of a cone is a truncated cone that has the cutting plane parallel to the base. The section made by the cutting plane is the upper base of the frustum; the perpendicular distance between its bases is the altitude of the frustum; and the portion of the slant height of the cone that is intercepted between the bases is the slant height of the frustum. Geom.-22 PROPOSITION IV. THEOREM. 690. The lateral area of a frustum of a cone of revolution is measured by the product of its slant height by half the sum of the circumferences of its bases. B Given S, the lateral surface, C and c, the circumferences of its bases, and L, the slant height of ABC-c, a frustum of a cone of revolution; To Prove: S is equal to L (C + c). Inscribe in the frustum ABC-c a frustum of a regular pyramid. If we denote its lateral surface by s, the perimeters of its upper and lower bases by p and P respectively, and its slant height by 1, whatever be the number of lateral faces of the pyramid. Conceive the number of lateral faces of the pyramidal frustum to be indefinitely increased by continually doubling the number of sides of its base polygons; then. since s, p, P, l, have for limits S, c, C, L, respectively, 691. COR. The lateral area of a frustum of a cone of revolution is measured by the product of its slant height by the circumference of a section equidistant from the bases. .. circumf. oa = (circumf. 04 + circumf. O'A'); .. circumf. oa × A A'= (circumf. 04+circumf. o'u̸') × AA'. PROPOSITION V. THEOREM. 692. The volume of a cone of revolution is measured by one third the product of its base by its altitude. Given: V, the volume, B, the base, H, the altitude, of a cone of revolution S-ABC; Inscribe in the cone a regular pyramid, S-ABC; then denoting its volume by v', its base by B', H being also its altitude, y' = B' X H, (555) whatever be the number of lateral faces of the pyramid. Conceive the number of lateral faces of the pyramid to be indefinitely increased; since y' and B' have for limits V and B resp., V = BX H. Q.E.D. (236) 693. COR. 1. If R denote the radius of the base; then, since B = π • R2, (398) 694. COR. 2. Similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of their radii. For if R and R' are the radii of two similar cones of revolution, H and H' their altitudes, V and ' their volumes, since the generating triangles are similar, (Hyp.) 695. SCHOLIUM. The volume of any cone is measured by one third the product of its base by its altitude. This may be proved by the same method as that employed in the proof of Prop. V., making the assumption before referred to (674). EXERCISE 837. If, by the method of Exercise 836, a straight line be found approximately equal to the circumference of a circle one yard in diameter, by what fraction of an inch will the line be too great, taking only two significant figures? 838. The diameters of the bases of a frustum of a cone are 10 in. and 8 in. respectively, and its slant height is 12 in. Find its lateral area. 839. Find the area of a section of that same cone equidistant from its bases. 840. Find the volume of a cone of revolution the radius of its base being 10 in. and its altitude 20 in. 841. What is the altitude of a similar cone of twice the volume ? PROPOSITION VI. THEOREM. 696. The volume of a frustum of a cone of revolution is measured by one third the product of its altitude by the sum of the bases of the frustum and a mean proportional between those bases. Given: V, the volume, B and B', the bases, and H, the altitude, of a frustum ABC-c; To Prove: V is equal to H (B + B' + √ B · B'). Inscribe in the frustum a frustum, ABC-c, of a regular pyramid; then, denoting its volume by v, its bases by b and b', H being its altitude, v = } H (b + b' + √b · b'), (559) whatever be the number of the lateral faces of the frustum. Conceive the number of lateral faces of the inscribed frustum to be indefinitely increased; since v, b, and b', have for limits V, B, and B', resp., V = }} H (B + B' + √ B · B'). Q.E.D. (236) 697. COR. If R and R' denote the radii of the bases of the frustum, as 698. SCHOLIUM. The volume of a frustum of any cone is measured by one third the product of its altitude, etc. The same remark applies here as in Arts. 674 and 695. |