Cone (geometry)
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- This article is about the geometric object, for other uses see Cone.
A cone is a three dimensional geometric shape. It is the locus of all line segments between a simply connected region of a plane (the base) and a point (the apex) outside the plane.
There are four types of cones; circular, elliptical, right and oblique all of which are conic solids. All pyramids are also cones, in otherwords it is the generalization of a pyramid to non-polygonal bases. In common usage and elementary geometry, a cone refers to a right circular cone. The boundary of a conic solid is a conic surface. A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum.
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[edit] Geometry
Cones may have a base of any shape and connect to an apex at any point outside the plane of the base. If the apex is located at right angle to the center of the base (i.e. a line joining the two is at right angles to the base plane), the cone is said to be a "right cone". Otherwise, it is called "oblique cone". A cone with a circular or elliptical base is called a circular cone or elliptical cone, respectively. If the base is a polyhedron, the cone is a pyramid.
[edit] Properties
The surface between the base and the apex is called the lateral surface. The total surface area is the sum of the lateral and base surface areas.
The line joining the center of the base and the apex is called an axis, the edge of the base is called a directrix and any line going through the apex and the directix is called a generatrix of the surface.
Every conic surface is ruled and developable.
A right circular cone with a the generatrix at angle <math>\theta</math> to the axis, has an aperture of <math>2\theta</math>.
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.
[edit] Formulas
The volume <math>V</math> of any conic solid is one third the area of the base <math>b</math> times the height <math>h</math> (the perpendicular distance from the base to the apex).
- <math>V = \frac{1}{3} b h </math>
The center of mass of a conic solid is at 1/4 of the height on the axis. For proof of this, see cone (geometry) proofs.
[edit] Right circular cone
For a circular cone with radius r, a more specific formula for volume is
- <math>V = \frac{1}{3} \pi r^2 h </math>
The surface area <math>A</math> is
- <math>A = \pi r (r + s)\,</math> where <math>s = \sqrt{r^2 + h^2}</math> is the slant height.
The first term in the area formula, <math>\pi r^2</math>, is the area of the base; while the second term, <math>\pi r s</math>, is the area of the curved side surface.
[edit] Conical surface
A conical surface <math>S</math> can be described parametrically as
- <math>S(t,u) = v + u q(t)</math>,
where <math>v</math> is the apex and <math>q</math> is the directrix.
A right circular cone whose axis is the <math>Z</math> coordinate axis, and whose apex is the origin, it is described parametrically as
- <math>S(t,u) = (u \cos\theta \cos t, u \cos\theta \sin t, u \sin\theta)</math>
and in implicit form by <math>S(x,y,z) = 0</math> where
- <math>S(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.</math>
More generally, a right circular cone with vertex at the origin, axis parallel to the vector <math>d</math>, and aperture <math>2\theta</math>, is given by the implicit vector equation <math>S(u) = 0</math> where
- <math>S(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2</math> or <math>S(u) = u \cdot d - |d| |u| \cos \theta</math>
where <math>u=(x,y,z)</math>, and <math>u \cdot d</math> denotes the dot product.
[edit] See also
ca:Con cs:Kužel de:Kegel (Geometrie) es:Cono (geometría) eo:Konuso eu:Kono (geometria) fr:Cône (géométrie) it:Cono (solido) he:חרוט lt:Kūgis nl:Kegel (ruimtelijk figuur) ja:円錐 no:Kjegle nn:Kjegle pl:Stożek (geometria) pt:Cone qu:Tuyru ru:Конус simple:Cone sk:Kužeľ fi:Kartio sv:Kon ta:கூம்பு tr:Koni zh:圆锥
