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Cone (geometry)

 

 
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Cone (geometry)
 
 
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A cone is a three-dimensional
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
 base and the surface (called the lateral surface) formed by the locus
Locus (mathematics)

In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve....
 of all straight line segments joining the apex to the perimeter
Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
 of the base.






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Cone 3d
A cone is a three-dimensional
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
 base and the surface (called the lateral surface) formed by the locus
Locus (mathematics)

In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve....
 of all straight line segments joining the apex to the perimeter
Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
 of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the lateral surface has a rotational symmetry
Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
.

In general, the base may be any shape, and the apex may lie anywhere (though it is often assumed that the base is bounded and has nonzero area, and that the apex lies outside the plane of the base). For example, a pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
 is technically a cone with a polygonal
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 base. In common usage in elementary geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, however, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 to its plane, and circular means that the base is a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.

Other mathematical meanings


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.

The boundary of an infinite or doubly infinite cone is a conical surface
Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight line 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....
, and the intersection of a plane with this surface is a conic section
Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
. For infinite cones, the word axis again usually refers to the axis of rotational symmetry (if any). One half of a double cone is called a nappe.

Depending on the context, "cone" may also mean specifically a convex cone
Convex cone

In linear algebra, a convex cone is a subset of a vector space that is closure under linear combinations with positive coefficients....
 or a projective cone
Projective cone

A projective cone in projective geometry is the union of all lines that intersect a projective subspace R and an arbitrary subset A of some other subspace S, disjoint from R....
.

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 of the lateral surface. (For the connection between this sense of the term "directrix" and the directrix of a conic section
Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
, see Dandelin spheres
Dandelin spheres

In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus:This concept is named in honor of Germinal Pierre Dandelin....
.)

The base radius of a circular cone is the radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 of its base; often this is simply called the radius of the cone. The aperture
Aperture

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 ray that come to a focus in the ....
 of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle ? to the axis, the aperture is 2?.

A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum
Frustum

A frustum is the portion of a solid?normally a Cone or pyramid ?which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen ....
. An elliptical cone is a cone with an elliptical
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 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
Visual hull

The Visual hull is a geometric entity created by shape-from-silhouette 3D reconstruction technique introduced by Laurentini.This technique assumes the foreground object in an image can be separated from...
).

Geometry


See also: Cone (geometry) proofs.


The volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
  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
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 of a conic solid of uniform density lies one-quarter of the way from the center of mass of the base to the vertex, on the straight line joining the two.

Right circular cone

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

For a right circular cone, the surface area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
  is   where     is the slant height
Slant height

The slant height of a right circular cone is the distance from any point on the circle to the apex of the cone.The slant height of a cone is given by the formula , where is the radius of the circle and is the height from the center of the circle to the apex of the cone....
. 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
Implicit function

In mathematics, an implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable....
 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 calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
 equation where   or   where , and denotes the dot product
Dot product

In mathematics, the dot product, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
.

See also

  • Cone (topology)
    Cone (topology)

    In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:of the product topology of X with the unit interval I = [0, 1]....
  • Democritus
    Democritus

    Democritus was an Ancient Greek philosopher born in Abdera in the north of Greece. He was the most prolific, and ultimately the most influential, of the pre-Socratic philosophers; his atomic theory may be regarded as the culmination of early Greek thought....
  • Pyramid (geometry)
    Pyramid (geometry)

    In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
  • Conic section
    Conic section

    File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
  • Quadric
    Quadric

    In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
  • Ruled surface
    Ruled surface

    In geometry, a surface is ruled if through every point of there is a straight line that lies on . The most familiar examples are the plane and the curved surface of a cylinder or cone ....
  • Hyperboloid
    Hyperboloid

    In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation  hyperboloid of one sheet,...
  • Ice cream cone
    Ice cream cone

    An ice cream cone or cornet is a cone-shaped pastry, usually made of a Wafer similar in texture to a waffle, in which ice cream is served, allowing it to be eaten without a bowl or spoon....


External links

from Math Is Fun
Math Is Fun

Math Is Fun is an educational website maintained by Rod Pierce devoted to the concept that mathematics is, indeed, fun.There are several aspects to the website:...
 from