Cone

A cone is a three-dimensional geometric shape formed by all straight lines connecting a circular base to a common point called the apex. The most common form is the right circular cone, where the apex lies directly above the base center. If the apex is offset from the base center, the cone is described as oblique. Cones can also be truncated by slicing parallel to the base, yielding a frustum.

The base has radius r and the height h is the perpendicular distance from base to apex. The slant height l is the distance from the apex to any point on the base edge, and for a right circular cone l = sqrt(r^2 + h^2). Key formulas: Volume V = (1/3) π r^2 h; lateral surface area A_l = π r l; total surface area A = π r (r + l).

Variants and cross-sections: A right circular cone is a special case; oblique cones tilt away from the vertical axis. A frustum is formed when a cone is truncated by a plane parallel to the base. Cross-sections perpendicular to the axis are circles whose radii vary linearly with height, reflecting the cone’s similarity along its axis.

Use and examples: The cone shape appears in everyday objects such as ice cream cones, traffic cones, and funnels. It is also relevant in engineering, optics, acoustics, and architectural design, where tapering parts or cone-like features influence performance and aesthetics.

Nature and naming: In botany, a cone refers to the seed-bearing structure of conifer trees (pinecones). These natural cones are named for their general conical form but are not mathematical cones.