![Cono 3D](https://upload.wikimedia.org/wikipedia/commons/5/51/Cono_3D.stl)
A cone is a
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the
apex
The apex is the highest point of something. The word may also refer to:
Arts and media Fictional entities
* Apex (comics), a teenaged super villainess in the Marvel Universe
* Ape-X, a super-intelligent ape in the Squadron Supreme universe
*Apex, ...
or
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
.
A cone is formed by a set of
line segments,
half-line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
s, or
lines connecting a common point, the apex, to all of the points on a base that is in a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
that does not contain the apex. Depending on the author, the base may be restricted to be a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, any one-dimensional
quadratic form in the plane, any closed
one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a
solid object; otherwise it is a
two-dimensional
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the ''lateral surface''; if the lateral surface is unbounded, it is a
conical surface.
In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a ''nappe''.
The
axis
An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to:
Mathematics
* Axis of rotation: see rotation around a fixed axis
* Axis (mathematics), a designator for a Cartesian-coordinat ...
of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a
circular symmetry.
In common usage in elementary
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, cones are assumed to be right circular, where ''circular'' means that the base is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
and ''right'' means that the axis passes through the centre of the base
at right angles to its plane.
If the cone is right circular the intersection of a plane with the lateral surface is a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
. In general, however, the base may be any shape
[Grünbaum, '' Convex Polytopes'', second edition, p. 23.] and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
, and that the apex lies outside the plane of the base). Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly.
A cone with a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
al base is called a
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
.
Depending on the context, "cone" may also mean specifically a
convex cone or a
projective cone.
Cones can also be generalized to
higher dimensions
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
.
Further terminology
The perimeter of the base of a cone is called the "directrix", and each of the line segments between the directrix and apex is a "generatrix" or "generating line" of the lateral surface. (For the connection between this sense of the term "directrix" and the
directrix of a conic section, see
Dandelin spheres.)
The "base radius" of a circular cone is the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of its base; often this is simply called the radius of the cone. The
aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane.
An ...
of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle ''θ'' to the axis, the aperture is 2''θ''.
![Acta Eruditorum - I geometria, 1734 – BEIC 13446956](https://upload.wikimedia.org/wikipedia/commons/6/68/Acta_Eruditorum_-_I_geometria%2C_1734_%E2%80%93_BEIC_13446956.jpg)
A cone with a region including its apex cut off by a plane is called a "
truncated cone"; if the truncation plane is parallel to the cone's base, it is called a
frustum
In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...
.
An "elliptical cone" is a cone with an
elliptical base.
A "generalized cone" is the surface created by the set of lines passing through a vertex and every point on a boundary (also see
visual hull
A visual hull is a geometric entity created by shape-from-silhouette 3D reconstruction technique introduced by A. Laurentini. This technique assumes the foreground object in an image can be separated from the background. Under this assumption, ...
).
Measurements and equations
Volume
The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of any conic solid is one third of the product of the area of the base
and the height
:
In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral
Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying
Cavalieri's principle – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the
method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
. This is essentially the content of
Hilbert's third problem
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...
– more precisely, not all polyhedral pyramids are ''scissors congruent'' (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.
Center of mass
The
center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
Right circular cone
Volume
For a circular cone with radius ''r'' and height ''h'', the base is a circle of area
and so the formula for volume becomes
:
Slant height
The slant height of a right circular cone is the distance from any point on the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
of its base to the apex via a line segment along the surface of the cone. It is given by
, where
is the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the base and
is the height. This can be proved by the
Pythagorean theorem.
Surface area
The
lateral surface area of a right circular cone is
where
is the radius of the circle at the bottom of the cone and
is the slant height of the cone.
The surface area of the bottom circle of a cone is the same as for any circle,
. Thus, the total surface area of a right circular cone can be expressed as each of the following:
*Radius and height
:
:(the area of the base plus the area of the lateral surface; the term
is the slant height)
:
:where
is the radius and
is the height.
*Radius and slant height
:
:
:where
is the radius and
is the slant height.
*Circumference and slant height
:
:
:where
is the circumference and
is the slant height.
*Apex angle and height
:
:where
is the apex angle and
is the height.
Circular sector
The
circular sector
A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the large ...
obtained by unfolding the surface of one nappe of the cone has:
*radius ''R''
:
*arc length ''L''
:
*central angle ''φ'' in radians
:
Equation form
The surface of a cone can be parameterized as
:
where
is the angle "around" the cone, and
is the "height" along the cone.
A right solid 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 function, 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
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
equation
where
:
or
where
, and
denotes the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
.
Elliptic cone
![Elliptical Cone Quadric](https://upload.wikimedia.org/wikipedia/commons/9/9b/Elliptical_Cone_Quadric.Png)
In the
Cartesian coordinate system, an ''elliptic cone'' is the locus of an equation of the form
:
It is an
affine image of the right-circular ''unit cone'' with equation
From the fact, that the affine image of a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
is a conic section of the same type (ellipse, parabola,...) one gets:
*Any ''plane section'' of an elliptic cone is a conic section.
Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see
circular section).
The intersection of an elliptic cone with a concentric sphere is a
spherical conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...
.
Projective geometry
![Australia Square building in George Street Sydney](https://upload.wikimedia.org/wikipedia/commons/8/8e/Australia_Square_building_in_George_Street_Sydney.jpg)
In
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, a
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
is simply a cone whose apex is at infinity. Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as
arctan
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
, in the limit forming a
right angle. This is useful in the definition of
degenerate conic
In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible variety, irreducible curve. This means that the defining equation is factorable over the comp ...
s, which require considering the
cylindrical conics.
According to
G. B. Halsted, a cone is generated similarly to a
Steiner conic
The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field.
The usual d ...
only with a projectivity and
axial pencils (not in perspective) rather than the projective ranges used for the Steiner conic:
"If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."
[ G. B. Halsted (1906) ''Synthetic Projective Geometry'', page 20]
Generalizations
The definition of a cone may be extended to higher dimensions; see
convex cone. In this case, one says that a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
''C'' in the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
R
''n'' is a cone (with apex at the origin) if for every vector ''x'' in ''C'' and every nonnegative real number ''a'', the vector ''ax'' is in ''C''.
In this context, the analogues of circular cones are not usually special; in fact one is often interested in ''
polyhedral cones''.
An even more general concept is the
topological cone, which is defined in arbitrary topological spaces.
See also
*
Bicone
In geometry, a bicone or dicone (from la, bi-, and Greek: ''di-'', both meaning "two") is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining ...
*
Cone (linear algebra)
*
Cylinder (geometry)
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infi ...
*
Democritus
Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek pre-Socratic philosopher from Abdera, primarily remembered today for his formulation of an atomic theory of the universe. No ...
*
Generalized conic In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of sums defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point wh ...
*
Hyperboloid
*
List of shapes
*
Pyrometric cone
Pyrometric cones are pyrometric devices that are used to gauge heatwork during the firing of ceramic materials. The cones, often used in sets of three, are positioned in a kiln with the wares to be fired and provide a visual indication of when t ...
*
Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
*
Rotation of axes
In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
*
Ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...
*
Translation of axes
In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...
Notes
References
*
External links
*
*
* {{MathWorld , urlname=GeneralizedCone , title=Generalized Cone
* An interactiv
Spinning Conefrom Maths Is Fun
Paper model coneCut a ConeAn interactive demonstration of the intersection of a cone with a plane
Elementary shapes
Surfaces