719 A right circular cone is a circular cone whose axis is perpendicular to its base. An oblique circular cone is a circular cone whose axis is oblique to its base. 720 A right circular cone is also called a cone of revolution, because it may be generated by the revolution of a right triangle about one of its legs as an axis. The hypotenuse generates the lateral surface, and the remaining leg generates the base of the cone. The hypotenuse in any position is an element of the surface, and any element is the slant height of the cone. 721 COROLLARY. All elements of a right circular cone are equal. 722 Similar cones of revolution are cones generated by the revolution of similar right triangles about homologous legs as axes. 723 A tangent line to a cone is an indefinite straight line which touches the lateral surface in one point only. 724 A tangent plane to a cone is an indefinite plane which contains an element of the cone without cutting its surface. The element contained by a tangent plane is called the element of contact. 725 A pyramid is inscribed in a cone when its lateral edges are elements of the cone and its base is inscribed in the base of the cone. 726 A pyramid is circumscribed about a cone when its lateral faces are tangent to the cone and its base is circumscribed about the base of the cone. 727 A truncated cone is the portion of a cone included between its base and a section formed by a plane cutting all its elements. 728 A frustum of a cone is a truncated cone in which the cutting section is parallel to the base. The lower base of the frustum is the base of the cone, the upper base of the frustum is the parallel section, and the altitude of the frustum is the perpendicular drawn between its bases. 729 The lateral surface of a frustum of a cone is the portion of the lateral surface of the cone included between the bases of the frustum. 730 The slant height of a frustum of a cone of revolution is the portion of any element of the cone included between the bases. PROPOSITION XXXIV. THEOREM 731 Every section of a cone made by a plane passing through its vertex is a triangle. HYPOTHESIS. S-ABC is a cone, and SBD is any section formed by a plane passing through the vertex S. CONCLUSION. The section SBD is a triangle. PROOF Draw the straight lines SB and SD. These straight lines lie in the lateral surface, and also in the plane SBD. § 716 § 511 That is, they are the lines in which the plane SBD cuts the lateral surface. .. the plane SBD cuts the lateral surface in the straight lines SB and SD. Also, BD is a straight line. .. the section SBD is a triangle. § 518 § 134 Q. E. D. EXERCISE 1242. What kind of triangle is the section made by a plane passing through the vertex of a cone of revolution ? 732 Every section of a circular cone made by a plane parallel to the base is a circle. S B HYPOTHESIS. The section abd of the circular cone S-ABD is parallel to the base ABD. CONCLUSION. The section abd is a circle. PROOF Let O be the center of the base, and let o be the point in which the axis SO pierces the parallel section. Through SO and any two elements SB and SC, pass planes cutting the base in the radii OB and OC, and the parallel section in the straight lines ob and oc. Since ob is to OB, and oc is I to OC, § 518 § 543 the Sob and SOB are similar; likewise A Soc and SOC. 733 COROLLARY. The axis of a circular cone passes through the center of every section parallel to the base. PROPOSITION XXXVI. THEOREM 734 If a pyramid whose base is a regular polygon is inscribed in or circumscribed about a circular cone, and if the number of sides of the base of the pyramid is indefinitely increased, The lateral area of the pyramid approaches the lateral area of the cone as a limit. The volume of the pyramid approaches the volume of the cone as a limit. Any section of the pyramid approaches the section of the cone made by the same plane as a limit. A A PROOF If the number of lateral faces of either pyramid is indefinitely increased, its base approaches the base of the cone as a limit. § 455 Therefore, the lateral area of the pyramid, its volume, and any section, approach as their respective limits the lateral area of the cone, its volume, and the section made by the same plane. Q. E. D. 735 COROLLARY. The above theorem remains true if for "pyramid" and "cone" we substitute "frustum of a pyramid" and "frustum of a cone" respectively. |