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Information about Cone (geometry)

This article is about the geometric object, for other uses see 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).

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.

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 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.

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).

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, though there is a reason they share a name: see dandelin spheres.)

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.

A pyramid is a special type of cone with a polygonal base.

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.

Formula

See also: Cone (geometry) proofs.


The volume of any conic solid is one third the area of the base times the height (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 is
  where     is the slant height.
The first term in the area formula, , is the area of the base, while the second term, , is the area of the lateral surface.

A right circular cone with height and aperture , whose axis is the coordinate axis and whose apex is the origin, is described parametrically as
where range over , , and , respectively.

In implicit form, the same solid is defined by the inequalities , where
.


More generally, a right circular cone with vertex at the origin, axis parallel to the vector , and aperture , is given by the implicit vector equation where
  or  
where , and denotes the dot product.

See also

External links

Cone (from the Greek κώνος, Latin conu) is a basic geometrical shape. It may also refer to:
  • Cone (software), a text-based e-mail client and news client for Unix-like operating systems.

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dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—i.e., length, width, and height or size and shape.
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Shape (OE. sceap Eng. created thing), refers to the external two-dimensional outline, appearance or configuration of some thing — in contrast to the matter or content or substance of which it is composed.
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line segment is a part of a line that is bounded by two end points, which have a finite length, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square.
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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. Geometry is one of the oldest sciences.
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circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
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perpendicular (or orthogonal) to each other if they form congruent adjacent angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
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plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
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A frustum (plural: frusta) is the portion of a solid – normally a cone or pyramid – which lies between two parallel planes cutting the solid. Degenerate cases are obtained for finite solids by cutting with a single plane only.
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locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve.
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perimeter is the distance around a given two-dimensional object. The word perimeter is a Greek root meaning measure around, or literally "around measure".

Practical uses

Perimeter and area play a great role in today's world.
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SET may stand for:
  • Sanlih Entertainment Television, a television channel in Taiwan
  • Secure electronic transaction, a protocol used for credit card processing,

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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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In linear algebra, a convex cone is a subset of a vector space that is closed under linear combinations with positive coefficients.

Definition

A subset C of a vector space V is a convex cone if and only if αx + βy
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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.
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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
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rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag (see opposite) has
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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. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their
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Dandelin spheres characterized thus:

Each Dandelin sphere touches, but does not cross, both the plane and the cone.


This concept is named in honor of Germinal Pierre Dandelin.
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Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area, although there are space-filling curves.
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plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
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circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
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ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
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pyramid is a polyhedron formed by connecting an n-sided polygonal base and a point, called the apex, by n triangular faces (n ≥ 3). In other words, it is a conic solid with polygonal base.
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POLYGONE is an Electronic Warfare Tactics Range located on the border between France and Germany. It is one of only two in Europe, the other being RAF Spadeadam.

The range, also referred to as the Multi-national Aircrew Electronic Warfare Tactics Facility (MAEWTF), is
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In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment. The radius is half the diameter.
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In optics, an aperture is a hole or an opening through which light is admitted. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane.
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Volume

  • Claim: The volume of a conic solid whose base has area b and whose height is h is .
Proof: Let be a simple planar loop in . Let be the vertex point, outside of the plane of .
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The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
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