the new edges continually bisecting the arcs of the base. Then p, s and I approach C, S and L respectively as their limits. mid, But however great the number of lateral faces of the pyra Also, since the area of the base is R2, the total area T of the cone is expressed by 660. COR. 2. Let S and S' denote the lateral areas of two similar cones of revolution, T and T their total areas, R and R' the radii of their bases, H and H' their altitudes, L and L' their slant heights. Since the generating triangles are similar, we have That is the lateral areas, or total areas, of similar cones of revolution are to each other as the squares of their slant heights, the squares of their altitudes, or the squares of the radii of their bases. PROPOSITION XXXV. THEOREM. 661. The lateral area of the frustum of a cone of revolution is equal to one-half the sum of the circumferences of its bases multiplied by the slant height. Let HBC-EFG be the frustum of a cone of revolution, and let S denote its lateral area, C and c the circumferences of its lower and upper bases, R and r the radii of the bases, and L the slant height. Inscribe in the frustum of the cone the frustum of the regular pyramid HBC-EFG, and denote the lateral area of this frustum by s, the perimeters of its lower and upper bases by P and p respectively, and its slant height by l. Then (P+p) l, = $570 (the lateral area of the frustum of a regular pyramid is equal to one-half the sum of the perimeters of its bases multiplied by the slant height). Now, let the number of lateral faces be indefinitely increased, the new elements constantly bisecting the arcs of the bases. Then P, p, and 1, approach C, c, and L, respectively as their limits. But, however great the number of lateral faces of the frustum of the pyramid, 662. COROLLARY. The lateral area of a frustum of a cone of revolution is equal to the circumference of a section equidistant from its bases multiplied by its slant height. For the section of the frustum equidistant from its bases cuts the frustum of the regular inscribed pyramid equidistant from its bases. Therefore the perimeter ILK = the sum of the perimeters HBC and E FG. § 142 And this will always be true, however great the number of the lateral faces of the frustum of the pyramid. Hence, circumference ILK = ences HBC and E FG. the sum of the circumfer$ 199 663. Any section of a cone parallel to the base is to the base as the square of the altitude of the part above the section is to the square of the altitude of the cone. A E D Let B denote the base of the cone, H its altitude, b a section of the cone parallel to the base, and h the altitude of the cone above the section. Let B' denote the base of an inscribed pyramid, b' the base of the pyramid formed in the section of the cone. Then B'b': H2: h2, § 566 (any section of a pyramid || to its base is to the base as the square of the from the vertex to the plane of the section is to the square of the altitude of the pyramid). Now let the number of lateral faces of the inscribed pyramid be indefinitely increased, the new edges continually bisecting the arcs in the base of the cone. Then B' and b' approach B and b respectively as their limits. mid, But however great the number of lateral faces of the pyra 664. The volume of any cone is equal to the product of one-third of its base by its altitude. Let V denote the volume, B the base, and H the al Let the volume of an inscribed pyramid A-C D E F G be denoted by V', and its base by B'. H will also be the altitude of this pyramid. Then V'B' X H, § 574 Now, let the number of lateral faces of the inscribed pyramid be indefinitely increased, the new edges continually bisecting the arcs in the base of the cone. Then V'approaches to V as its limit, and B' to B as its limit. 665. COROLLARY 1. If the cone be a cone of revolution, and R be the radius of the base, then B = π k2 (§ 381); . . V = } π R2 × H. 666. COR. 2. Similar cones of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases. For, let R and R' be the radii of two similar cones of revolution, H and H' their altitudes, V and V' their volumes. Since the generating triangles are similar, we have |