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Encyclopedia > Cone (geometry)
A right circular cone and an oblique cone
A right circular cone and an oblique cone

A cone is a three-dimensional geometric shape consisting of all line segments joining a single point (the apex or vertex) to every point of a two-dimensional figure (the base). Look up cone in Wiktionary, the free dictionary. ... Image File history File links Download high-resolution version (1024x768, 439 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Cone (geometry) User:DemonDeLuxe/Graphics ... Image File history File links Download high-resolution version (1024x768, 439 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Cone (geometry) User:DemonDeLuxe/Graphics ... 2-dimensional renderings (ie. ... In geometry, two sets of points are of the same shape precisely if one can be transformed to another by dilating (i. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...


The axis of a cone is the line joining the apex to the center of the base (suitably defined). In common usage and in elementary geometry, the base is a circle, and the axis is perpendicular to the plane of the base, i.e. cones are assumed to be right circular. A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum. Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Circle illustration This article is about the shape and mathematical concept of circle. ... Fig. ... This article is about the mathematical construct. ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ...


The term "cone" sometimes refers just to the lateral surface of a solid cone, the locus of all line segments that join the apex to the perimeter of the base. In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... The perimeter is the distance around a given two-dimensional object. ...


In mathematical usage, the word "cone" is used also for an infinite cone, the union of any set of half-lines that start at a common apex point. This kind of cone does not have a bounding base, and extends to infinity. A doubly infinite cone, or double cone, is the union of any set of straight lines that pass through a common apex point, and therefore extends symmetrically on both sides of the apex. Depending on the context, the word may also mean specifically a convex cone or a projective cone. This article is about sets in mathematics. ... In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In linear algebra, a convex cone is a subset of a vector space that is closed under linear combinations with positive coefficients. ... A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R. In the special case that R is a single...


The boundary of an infinite or doubly infinite cone is a conical surface. For infinite cones, the word axis usually refers to the axis of rotational symmetry (if any). In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... The triskelion appearing on the Isle of Man flag. ...

Contents

Elements and special cases

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.) Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...


In general, the base of a cone may have any shape, and the apex may lie anywhere. However, it is often assumed that the base is bounded and has nonzero area, and that the apex lies outside the plane of the base. Circular cones and elliptical cones have, respectively, circular and elliptical bases. 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. Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... This article is about the mathematical construct. ... Circle illustration This article is about the shape and mathematical concept of circle. ... For other uses, see Ellipse (disambiguation). ...


A pyramid is a special type of cone with a polygonal base. This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... Look up polygon in Wiktionary, the free dictionary. ...


The base radius of a circular cone is the radius of its base; often this is simply called the radius of the cone. The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes and angle θ to the axis, the aperture is 2θ. Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... a big (1) and a small (2) aperture For other uses, see Aperture (disambiguation). ...


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). Claim: The volume of a conic solid whose base has area b and whose height is h is . ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

V = frac{1}{3} b h

The center of mass of a conic solid is at 1/4 of the height on the axis. In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...


Right circular cone

For a circular cone with radius r and height h, the formula for volume becomes

V = frac{1}{3} pi r^2 h.

The surface area A is This article is about the physical quantity. ...

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, πr2, is the area of the base, while the second term, πrs, is the area of the lateral surface. Slant Height of a Right Circular cone is the Distance from any point on the Circle to the Apex of the Cone. ...


A right circular cone with height h and aperture , whose axis is the Z coordinate axis and whose apex is the origin, is described parametrically as

S(s,t,u) = (utanscost,utanssint,u)

where s,t,u range over [0,θ), [0,2π), and [0,h], respectively.


In implicit form, the same solid is defined by the inequalities { S(x,y,z) leq 0, zgeq 0, zleq h}, where In mathematics, to give a function implicitly is to give an equation that at least in part has the same graph as . ...

S(x,y,z) = (x^2 + y^2)(costheta)^2 - z^2 (sin theta)^2,.

More generally, a right circular cone with vertex at the origin, axis parallel to the vector d, and aperture , is given by the implicit vector equation S(u) = 0 where Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

S(u) = (u cdot d)^2 - (d cdot d) (u cdot u) (cos theta)^2   or   S(u) = u cdot d - |d| |u| cos theta

where u = (x,y,z), and u cdot d denotes the dot product. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...


See also

In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse one end of the cylinder to a point. ... This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation). ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... In geometry, a surface is ruled if through every point of there is a straight line that lies on . ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, a = b, it is a hyperboloid of revolution. ...

External links


  Results from FactBites:
 
Cone - Wikipedia, the free encyclopedia (459 words)
A volcanic cone is a mountain formed by material ejected from a volcanic vent.
In vertebrate anatomy, a cone cell is a type of light-sensitive cell found along with rods in the retina of the eye.
In ceramics manufacture, a pyrometric cone is a conical or pyramidal mineral stick that is used to gauge the temperature of an oven.
  More results at FactBites »

 
 

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