Britannica Sources used in this answer

Which conic section is formed when a plane cuts one side of a cone?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

When a plane intersects a single nappe of a cone, the resulting conic section is a parabola if the plane is parallel to an element (a side) of the cone.1

More generally, the specific conic section formed depends on the angle of the plane relative to the cone's axis. The four primary types of conic sections are:1

Circle: Formed when the plane is perpendicular to the axis of the cone.1
Ellipse: Formed when the plane cuts through one nappe at an angle that is not parallel to an element of the cone.12
Parabola: Formed when the plane is parallel to an element of the cone.1
Hyperbola: Formed when the plane cuts through both nappes of a double cone.1

Degenerate cases, such as a single point or a line, can also occur if the plane passes through the apex of the cone.1