150. Tangent Plane. A plane which contains an element of a cone but does not cut the surface (as in the first figure below) is called a tangent plane. From this definition it is evident that A plane passing through a tangent to the base of a circular come and the element through the point of contact is tangent to the come. If a plane is tangent to a circular come, its intersection with the plane of the base is tangent to the base. 151. Inscribed Pyramid. A pyramid whose lateral edges are elements of a cone and whose base is inscribed in the base of the cone is called an inscribed pyramid. The second figure above shows an inscribed pyramid. The cone is said to be circumscribed about the pyramid. 152. Circumscribed Pyramid. A pyramid whose lateral faces are tangent to the lateral surface of a cone and whose base is circumscribed about the base of the cone is called a circumscribed pyramid. The third figure above shows a circumscribed pyramid. The cone is said to be inscribed in the pyramid. $$ 150-154 CONE AS A LIMIT 107 153. Frustum of a Cone. The portion of a cone included between the base and a section parallel to the base is called a frustum of a cone. This figure shows a frustum of a cone of revolution. The base of the cone and the parallel section are together called the bases of the frustum. The terms altitude and lateral area as applied to a frustum of a cone, and slant height as applied to a frustum of a right circular cone, have the same meaning as they do when applied to a frustum of a pyramid. As with frustums of regular pyramids, only frustums of cones of revolution have a slant height. 154. Cones and Frustums as Limits. The following properties of cones and frustums, similar to those given in $ 139, may be assumed without proof from a study of the figures accompanying the statements: 1. If a pyramid whose base is a regular polygon is inscribed in or circumscribed about a circular cone, and if the number of lateral faces of the pyramid is indefinitely increased, the volume of the cone is the limit of the volume of the pyramid, and the lateral area of the cone is the limit of the lateral area of the pyramid. 2. The volume of a frustum of a cone is the limit of the volumes of the corresponding frustums of the inscribed and circumscribed pyramids whose bases are regular polygons as the number of lateral faces is indefinitely increased, and the lateral area of the frustum of a cone is the limit of the lateral areas of the corresponding frustums of the inscribed and circumscribed pyramids. Proposition 21. Lateral Area of a Cone 155. Theorem. The lateral area of a cone of revolution is half the product of the slant height and the circumference of the base. Given a cone of revolution; S, the lateral area; C, the circumference of the base; and 1, the slant height. Prove that S=C. Proof. In a regular circumscribed pyramid whose lateral area is L and whose perimeter of base is p, L=] lp. $ 118 If the number of lateral faces is indefinitely increased, L+S, § 154, 1 and PC. $ 29,1 Then i lp →ic. $ 28, 1 ::S=ic. $ 28, 2 156. Corollary. In a cone of revolution of lateral area S, total area T, slant height l, and radius r, S=mrl, and T=mr(1+r). 157. Corollary. The theorem of § 120 applies to a circular cone. The necessary changes in wording are obvious. $$ 155-159 LATERAL AREA 109 Proposition 22. Lateral Area of a Frustum 158. Theorem. The lateral area of a frustum of a cone of revolution is half the product of the slant height and the sum of the circumferences of its bases. Given a frustum of a cone of revolution; S, the lateral area; Cand C', the circumferences of the bases; and l, the slant height. Prove that S=žl(C+C'). Proof. Let L be the lateral area of the corresponding frustum of a regular circumscribed pyramid, and let p, p' be the perimeters of the bases corresponding to C, C' respectively. Then L=1(p+p'). $ 119 If the number of lateral faces is indefinitely increased, L+S. $ 154, 2 From $ 29, 1, we may assume that p+p'+C+C'. Then žl(p+p') }(C+C'). $ 28, 1 ..S=il(C+C'). $ 28, 2 159. Corollary. The lateral area of a frustum of a cone of revolution is the product of the slant height and the circumference of a section equidistant from the bases. Proposition 23. Volume of a Cone 160. Theorem. The volume of a circular cone is one third the product of the base and the altitude. Given a circular cone; V, the volume; B, the area of the base; and h, the altitude. Prove that V=jBh. Proof. Let a pyramid whose base is a regular polygon be inscribed in the cone. Let V' be the volume and B' the area of the base of the inscribed pyramid. Then V'=j B'h. $ 124 If the number of lateral faces is indefinitely increased, VV, $ 154, 1 and B'→B. $ 29, 2 Then B'h + }Bh. $ 28, 1 ...V=ž Bh. $ 28, 2 161. Corollary. In a circular cone of volume V, radius r, and altitude h, V=ž trh. |