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What shape is formed by the intersection of a cone and a plane that is parallel to a cone's base not containing the cone's vertex?
The shape described by the intersection of the cone and the plane is simply a circle.
If a right circular cone is intersected by a plane so that the intersection goes through the cone's vertex as well as an edge of each nappe what shape is produced?
If a right circular cone is intersected by a plane so that the intersection goes through the cone's vertex as well as an edge of each nappe, the shape produced is a line. Not asked, but... If the angle of the plane is less than the angle of the cone, then the intersection is a point. If the angle of the plane is greater than the angle of the cone, then the intersection is two lines intersecting at the vertex. If the plane insersects at other than the vertex, then the intersection is a circle when the plane is perpendicular to the cone's axis, an ellipse when the plane's angle is less than the cone's angle, a parabola when the planes's angle equals the cone's angle, and two hyperbole's in the last case.
What term best describes the point line or curve defined by the intersection of a cone and a plane?
Conic section
If a right circular cone is intersected by a plane only at its vertex what shape is produced?
The intersection will consist of only one point.
What figure is formed by the intersection of a cone?
The most interesting intersections of cone are its planar sections (intersections with a plane). These sections are called "conic" The figures can be 1) dot - for plane going through apex 2) two lines - for plane containing cone axis 3) one line - for plane going through apex and touching the cone 3) circle - for plane orthogonal to cone axis 4) ellipsis - for plane that intersects the cone axis and generating lines 5) parabola - for plane parallel to a generating line 4) hyperbola - other cases See wikipedia's "conic section" article
