672. DEF. Similar cones of revolution are cones generated by the revolution of similar right triangles about homologous arms. 673. DEF. A line is tangent to a cone, if the line touches the cone in one point only and does not intersect it if produced. 674. DEF. A plane is tangent to a cone, if it contains an element but no other point of the cone. 675. DEF. A pyramid is inscribed in a cone when its base is inscribed in the base of the cone and its vertex coincides with the vertex of the cone. 676. A pyramid is circumscribed about a cone when its base is circumscribed about the base of the cone and its vertex coincides with the vertex of the cone. 677. DEF. A frustum of a cone is the portion included between its base and a plane parallel to the base. The lower base of the frustum is the base of the cone, and the upper base is the section made by the plane. 678. If pyramids are circumscribed about and inscribed in a cone with volume V, lateral surface L, base B, circumference of the base C, and the number of lateral faces of the pyramids is increased indefinitely, then L is the limit of the lateral surfaces of the circumscribed pyramids. V is the limit of the volumes of the inscribed pyramids. B is the limit of the base of the inscribed pyramid. c is the limit of the perimeter of the base of the circumscribed pyramid. NOTE. The first two statements may be considered as definitions; the last two are assumptions. It will be further assumed in the following section that (a) The limit of the sum of several variables equals the sum of the limits of these variables. (b) The limit of the product of two variables equals the product of their respective limits, when no one of these limits is zero. All four assumptions are capable of proof. PROPOSITION XXVIII. THEOREM 679. Every section of a cone made by a plane passing through its vertex is a triangle. Given ABC, a section of the cone made by plane passing through vertex A. .. this line is the intersection of the conical surface and the given plane. In like manner it follows that the straight line AC is the intersection of conical surface and the given plane. BC is a straight line. .. ABC, the section, is a triangle. (480) Q. E. D. 680. COR. Every section of a right cone made by a plane passing through its vertex is an isosceles triangle. PROPOSITION XXIX. THEOREM 681. Every section of a circular cone made by a plane parallel to the base is a circle. Q Α B Given A'B'C', a section of cone V-ABC made by a plane || ABC, O the center of the base, and o', the point of intersection of vo and plane A'B'C'. To prove. A'B'C' is a circle. Proof. Pass planes through vo and OB, and through vo and oc, and let them intersect plane A'B'C' in O'B' and o'c' respectively. 682. COR. 1. The axis of a circular cone passes through the center of every section which is parallel to the base, or The locus of the centers of the sections of a circular cone made by planes parallel to the base is the axis of the cone. 683. COR. 2. Sections made by planes parallel to the bases of a circular cone are to each other as the squares of their radii, or as the squares of their distances from the vertex of the cone. PROPOSITION XXX. THEOREM 684. The lateral area of a cone of revolution is equal to half the product of the slant height by the circumference of the base. Given Z lateral area, C the circumference of the base, and S the slant height of the cone; L' the lateral area, P the perimeter of the regular polygon forming the base of a circumscribed pyramid. HINT. Circumscribe a pyramid; its slant height is S. Use Theorem of Limits. 685. COR. If L is the lateral area, T the total area, H the altitude, s the slant height, R the radius of the base, of a cone of revolution, L=TRS. TTR (SR). Ex. 1599. Find the lateral area of a cone of revolution whose radius is 12 and whose altitude is 5. Ex. 1600. Find the lateral area of a cone of revolution if the hypotenuse of the generating triangle be 10 in. and the acute angles be 45° each. Ex. 1601. Find the altitude of a cone of revolution if L = R = 4. 36 π, and |