647. DEF. A circular cone is a cone whose base is a circle; the axis of a circular cone is the straight line joining the vertex and the center of the base.

648. DEF. A right circular cone is one whose axis is perpendicular to the base; an oblique circular cone is one whose axis is not perpendicular to the base.

NOTE. In this book only circular cones are treated of.

649. DEF. A cone of revolution is a right circular cone, for this latter may be generated by a right triangle revolving about one of its arms as an axis.

650. DEF. The altitude of a cone is the perpendicular distance from the vertex to the plane of the base.

651. DEF. Similar cones of revolution are cones generated by the revolution of similar right triangles about homologous arms. 652. DEF. A tangent line to a cone is a line which touches the cone in one point only and does not intersect it.

653. DEF. A tangent plane to a cone is a plane which contains one element of the cone and but one, and does not intersect the cone.

654. 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.

655. 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.

656. 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.

PROPOSITION XXIX. THEOREM

657. Every section of a cone made by a plane passing through its vertex is a triangle.

Hyp. ABC is a section of the cone made by plane passing through vertex A.

To prove

ABC a triangle.

Proof.

BC is a straight line.

(Why?)

AB and AC are elements of the conical surface.

.. AB and AC are straight lines.

AB and AC are in the cutting plane.

(Why?)

.. they are intersections of the plane and the conical surface.

.. ABC, the section, is a triangle.

(Why?)

Q.E.D.

658. COR. Every section of a right cone made by a plane passing through its vertex is an isosceles triangle.

PROPOSITION XXX. THEOREM

659. Every section of a circular cone made by a plane parallel to the base is a circle.

D'

A

B

Hyp. A'B'C'D' is a section of cone V-ABCD made by a plane || to ABCD, O the center of the base, O' the point where axis VO intersects the section. OA, OD, and O'A', O'D' are intersections of a plane through V and any element VA with ABCD and A'B'C'D' respectively.

Proof. O'A' and O'D' are || to OA and OD respectively.

660. 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.

661. 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 XXXI. THEOREM

662. 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

Hyp. S is lateral area, C the circumference of the base, and L the slant height of the cone; S' the lateral area, P the perimeter of the regular polygon forming the base of a circumscribed pyramid.

To prove

S = C × L.

HINT. Circumscribe a pyramid; its slant height is L. Use Theorem of Limits.

663. COR. If S is the lateral area, T the total area, H the altitude, L the slant height, R the radius of the base, of a cone of revolution,

S=TRL.

T=TR(L+R).

Ex. 1130. Find the lateral area of a cone of revolution if the hypotenuse of the generating triangle be 10 inches and the acute angles be 45° each.