Cone

In common speaking and geometry, a cone is a solid object that one gets when one rotates a right triangle around one of its two short sides, the cone's axis. The disk made by the other short side is called the base, and the point of the axis which is not on the base is the cone's apex or vertex. An object that is shaped like a cone is conical.

In more technical terms, a cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the lateral surface; if the lateral surface is unbounded, it is a conical surface.

In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe. The volume ${\displaystyle }$ of a cone is one third of the product of the area of the base ${\displaystyle A_{B}}$ and the height ${\displaystyle h}$^[1]

${\displaystyle V={\frac {1}{3}}A_{B}h.}$

1. ↑ Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01). Elementary Geometry for College Students. Cengage Learning. ISBN 9781285965901.