Cone

A cone is a three-dimensional geometric shape. It consists of a base—usually circular—and a single point extending beyond the plane of the base, called the apex. The apex and the circular base are connected by a curved surface (lateral surface).

When the line from the apex to the center of the base is perpendicular to the base, it is called a right circular cone. If it is slanted, it is an oblique cone. In general geometry, a cone can be defined by drawing lines from a single point through every point on a closed curve.

Measurements and properties

The basic measurements are the radius of the base (r), the perpendicular height from apex to base (h), and the slant height along the lateral surface (l), which for a right circular cone is found using the Pythagorean theorem: l equals the square root of (r² + h²). The volume of a cone is one-third the volume of a cylinder with the same base and height, which is (1/3)·π·r²·h. The lateral surface area of a right circular cone is π·r·l, and the total surface area includes the base area, π·r².

Frustum and conic sections

When a cone is cut by a plane parallel to the base, the remaining lower portion is called a frustum. The intersections of a plane with a cone produce shapes known as conic sections: a circle, ellipse, parabola, and hyperbola. The type depends on the angle of the cutting plane relative to the cone's lateral surface.

Applications

The cone shape appears in engineering and everyday life: road signs, sound funnels, hair dryers, and parts of machinery. It also appears in advanced mathematics, particularly in the definition of a cone in vector space, where it refers to a set closed under multiplication by positive scalars.