In elementary work, a cone usually consists of a circle as base, a vertex lying directly above the centre of the circle, and the curved surface formed by the line segments joining the vertex to the points of the circle. The distance from the vertex to the centre of the base is the height, and the length of any of the line segments is the slant height. For a cone with base of radius r, height h and slant height l, the volume equals ⅓πr2h and the area of the curved surface equals πrl.
In more advanced work, a cone is the surface consisting of the points of the lines, called generators, drawn through a fixed point V, the vertex, and the points of a fixed curve, the generators being extended indefinitely in both directions. Then a right-circular cone is a cone in which the fixed curve is a circle and the vertex V lies on the line through the centre of the circle and perpendicular to the plane of the circle. The axis of a right-circular cone is the line through V and the centre of the circle, and is perpendicular to the plane of the circle. All the generators make the same angle with the axis; this is the semi-vertical angle of the cone. The right-circular cone with vertex at the origin, the z-axis as its axis, and semi-vertical angle α, has equation x2+y2=z2 tan2α. See also quadric cone.