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, halflines, 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 onedimensional quadratic form in the plane, any closed onedimensional 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 twodimensional object in threedimensional 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 halflines, 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 of a cone is one third of the product of the area of the base and the height
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In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.