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. through vertex A. To prove Given ABC, a section of the cone made by plane passing ABC a triangle. Proof. Join A and B by a straight line. This line is an element of the cone. (662) This line lies in the given plane. (478) ... 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. 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. A Given L 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 = πR. S. T = πR (S + R). 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 = 36 π., and R = 4. PROPOSITION XXXI. THEOREM 686. The volume of a cone is equal to one third the product of its base by its altitude. Given the volume, B the base, and H the altitude of the cone; v' the volume, B' the regular polygon forming the base of an inscribed pyramid. 687. COR. If the cone is a cone of revolution, with R as radius of B, then V = πR2H. Ex. 1602. Find the volume of a cone of revolution with radius 6 and altitude 2. Ex. 1603. Find the volume of a cone of revolution whose radius is 5 and whose slant height is 13. Ex. 1604. Find the lateral surface of a cone of revolution if its volume is 314 and its altitude is 3. (Assume ᅲ = 3.14.) Ex. 1605. The axis of a circular cone equals 17 in., its projection upon the base equals 8 in. Find radius of the base if the volume equals 80 π. PROPOSITION XXXII. THEOREM 688. The lateral areas, or the total areas, of two similar cones of revolution are to each other as the squares of their altitudes, as the squares of their radii, |