CONE (GEOMETRY)

A 'cone' is a three-dimensional geometric shape bounded by a simply connected region of a plane (the ''base'') and a surface (the ''lateral surface'') described by the locus of all line segments joining the perimeter of the base to a point (the ''apex'' or ''vertex'') lying off the plane of the base. In common usage in elementary geometry, "cone" usually means a right circular cone (see below).
Sometimes the term "cone" refers just to the boundary, or surface, of such a solid (also known as a ''conic surface''). In mathematical usage, the straight lines that generate the lateral surface are often considered to be indefinitely extended in both directions; thus the cone is not delimited by a base, and extends symmetrically on both sides of the apex. Such a cone (sometimes called a ''double cone'') consists of two halves joined at the apex, each of which is known as a ''nappe''.
The line joining the apex and the center of the base, suitably defined, is called the ''axis''. The perimeter of the base is called the ''directrix'', and each of the line segments between the directrix and apex is a ''generatrix'' of the lateral surface. (The term "directrix" here should not be confused with its meaning as the generator of a conic section.) A cone with its apex cut off by a plane parallel to its base is called a truncated cone or ''frustum''.
In general, cones may have a base of any shape, and the apex may lie anywhere outside the plane of the base. ''Circular cones'' and ''elliptical cones'' have, respectively, circular and elliptical bases. A ''pyramid'' is a special type of cone with a polygonal base. If the axis of the cone is at right angles to its base then it is said to be a ''right cone'', otherwise it is an ''oblique cone''.

Contents

Properties

Formulae

Right circular cone

Conical surface

See also

External links

Properties

Every conic surface is ruled and developable.
A right circular cone with a generatrix at angle ''θ'' to the axis has an aperture of 2''θ''.
Mathematically, an elliptical conic surface is a special case of a conic section which is formed by a "conical quadric", which is a special case of a quadric.
A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry there is no difference between the cylindrical and conical surfaces, and the two halves of the latter become a single connected surface.

Formulae

:''See also: Cone (geometry) proofs.''
The volume $V$ of any conic solid is one third the area of the base $b$ times the height $h$ (the perpendicular distance from the base to the apex).
:
The center of mass of a conic solid is at 1/4 of the height on the axis.

Right circular cone

For a circular cone with radius ''r'' and height ''h'', the formula for volume becomes
:
The surface area $A$ is
:$A\; =pi\; r^2\; +\; pi\; r\; s,$   where   $s\; =\; sqrt\{r^2\; +\; h^2\}$   is the slant height.
The first term in the area formula, $pi\; r^2$, is the area of the base, while the second term, $pi\; r\; s$, is the area of the lateral surface.

Conical surface

A conical surface $S$ can be described parametrically as
:$S(t,u)\; =\; v\; +\; u\; q(t),$,
where $v$ is the apex and $q$ is the directrix.
A right circular cone whose axis is the $Z$ coordinate axis, whose apex is the origin, and whose aperture is $2\; heta$, is described parametrically as
:$S(t,u)\; =\; (u\; cos\; heta\; cos\; t,\; u\; cos\; heta\; sin\; t,\; u\; sin\; heta,)$
and in implicit form by $S(x,y,z)\; =\; 0$ where
:$S(x,y,z)\; =\; (x^2\; +\; y^2)(cos\; heta)^2\; -\; z^2\; (sin\; heta)^2,$.
More generally, a right circular cone with vertex at the origin, axis parallel to the vector $d$, and aperture $2\; heta$, is given by the implicit vector equation $S(u)\; =\; 0$ where
:$S(u)\; =\; (u\; cdot\; d)^2\; -\; (d\; cdot\; d)\; (u\; cdot\; u)\; (cos\; heta)^2$   or   $S(u)\; =\; u\; cdot\; d\; -\; |d|\; |u|\; cos\; heta$
where $u=(x,y,z)$, and $u\; cdot\; d$ denotes the dot product.

See also

★ Cone (topology)

★ Pyramid (geometry)

★ Conic section

★ Quadric

★ Ruled surface

★ Hyperboloid

External links

★ Spinning Cone from Math Is Fun

★ Paper model cone