Cone (geometry)

# Cone (geometry)

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A cone is an -dimensional
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...

geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or vertex. Formally, it is the solid figure formed by the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

of all straight line segments that join the apex to the base. The term "cone" is sometimes used to refer to the surface or the lateral surface of this solid figure (the lateral surface of a cone is equal to the surface minus the base).

The axis of a cone is the straight line (if any), passing through the apex, about which the base has a rotational symmetry
Rotational symmetry
Generally speaking, an object with 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 has...

.

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 ....

, 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 lines or planes are considered perpendicular to each other if they form congruent adjacent 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 points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base. In general, however, 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. It is a conic solid with polygonal base....

is technically a cone with a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

al base.

## Other mathematical meanings

In mathematical usage, the word "cone" is used also for an infinite cone, the union of a set of half-lines that start at a common apex point and go through a base. Observe that an infinite cone is not bounded by its base and extends to infinity. A doubly infinite cone, or double cone, is the union of a set of straight lines that pass through a common apex point and go through a base, therefore double infinite cones extend 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 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...

, and the intersection of a plane with this surface is a conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

. For infinite cones, the word axis again usually refers to the axis of rotational symmetry (if any). Either half of a double cone on one side of the apex 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 over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

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....

.

Also, a oblique shaped cone is known as a circular based pyramid due to the inwardly converging hypos towards the apex point.

## 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, see Dandelin spheres
Dandelin spheres
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin...

.)

In classical geometry, a radius 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, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

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 travels. 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. The aperture determines how collimated the admitted rays are,...

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 2θ.

A cone with 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
Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

. An elliptical cone is a cone with an elliptical
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

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 fromthe background...

).

### Surface Area

The lateral surface
Lateral surface
In geometry, the lateral surface of a solid is the face or surface of the solid on its sides. That is, any face or surface that is not a base.-Sources:*...

area of a right circular cone is where is the radius of the circle at the bottom of the cone and is the lateral height of the cone (given by the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

where is the 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 is:
or

### Volume

The volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of any conic solid is one third of the product of the area of the base and the height (the perpendicular distance from the base to the apex).

In modern math, 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
Cavalieri's principle
In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...

– 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
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 area between the n-th polygon and the containing shape will...

. This is essentially the content of Hilbert's third problem
Hilbert's third problem
The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. 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 many polyhedral pieces which can be reassembled to yield the...

– 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
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

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

For a circular cone with radius R and height H, the formula for volume becomes

where r is the radius of the cone at height h:

Thus:

Thus:

For a right circular cone, the surface area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

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
Implicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...

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
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

.

## Projective geometry

In projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

, a cylinder is simply a cone whose apex
Apex (geometry)
In geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs.*In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side....

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 the limit forming a right angle
Right angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...

.

This is useful in the definition of degenerate conic
Degenerate conic
In mathematics, a degenerate conic is a conic that fails to be an irreducible curve...

s, which require considering the cylindrical conics.

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

• Democritus
Democritus
Democritus was an Ancient Greek philosopher born in Abdera, Thrace, Greece. He was an influential pre-Socratic philosopher and pupil of Leucippus, who formulated an atomic theory for the cosmos....