|The right triangle ABC with right angle at A is rotated about the side AC to produce a cone. The axis of the cone is AC, and its base is the circle with center at A and radius AB.
The three different kinds of cone are not used by Euclid in the Elements, but they were important in the theory of conic sections until Apollonius’ work Conics. In Euclid’s time conic sections were taken as the intersections of a plane at right angles to an edge (straight line from the vertex) of a cone. When the cone is acute-angled, the section is an ellipse; when right-angled, a parabola; and when obtuse-angle, a hyperbola. Even the names of these three curves were given by the kind of angle, so, for instance, Euclid knew a parabola as a “section of a right-angled cone.” It was Apollonius who named them ellipse, parabola, and hyperbola.