Cone (geometry)

A geometric cone joins a fixed point to points of a plane curve by straight lines. A circular cone uses a circle as that curve; its solid version includes the enclosed circular cross-sections. For a right circular cone of height $h$ and base radius $R$, the cross-section radius varies linearly along the axis, and its volume is $\pi R^{2}h/3$ by integration. An elliptic cone allows elliptical cross-sections. The construction differs from the abstract topological cone, which does not specify lengths or angles.

For a coaxial cylinder standing on the base of a right circular cone of height $h>0$ and semi-vertical angle $0<\alpha <\pi /2$, similar triangles give radius $r=(h-x)\tan \alpha$ at cylinder height $x$. Its volume is $V=\pi x(h-x{)}^2{\tan }^2\alpha$. Differentiation gives $V^{\prime }=\pi {\tan }^2\alpha (h-x)(h-3x)$, so the unique positive-volume maximum has $x=h/3$ and occupies $4/9$ of the cone's volume. Its curved surface area is instead $A=2\pi x(h-x)\tan \alpha$, maximized at $x=h/2$. Changing the optimized quantity changes the optimal shape, even though the geometric constraint is identical.