A cone is formed by a set ofline segments,half-lines, orlines connecting a common point, the apex, to all of the points on a base. 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 adouble cone. Each of the two halves of a double cone split at the apex is called anappe.
Depending on the author, the base may be restricted to a circle, any one-dimensionalquadratic form in the plane, any closedone-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is asolid object; otherwise it is anopen surface, atwo-dimensional object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called thelateral surface; if the lateral surface isunbounded, it is aconical surface.
Theaxis of a cone is the straight line passing through the apex about which the cone has acircular symmetry.In common usage in elementary geometry, cones are assumed to beright circular, i.e., with a circle baseperpendicular to the axis.[1] If the cone is right circular the intersection of a plane with the lateral surface is aconic section. In general, however, the base may be any shape[2] and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finitearea, and that the apex lies outside the plane of the base). Contrasted with right cones areoblique cones, in which the axis passes through the centre of the base non-perpendicularly.[3]
The perimeter of the base of a cone is called thedirectrix, and each of the line segments between the directrix and apex is ageneratrix orgenerating line of the lateral surface. (For the connection between this sense of the termdirectrix and thedirectrix of a conic section, seeDandelin spheres.)
Thebase radius of a circular cone is theradius of its base; often this is simply called the radius of the cone.Theaperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angleθ to the axis, the aperture is 2θ. Inoptics, the angleθ is called thehalf-angle of the cone, to distinguish it from the aperture.
Illustration fromProblemata mathematica... published inActa Eruditorum, 1734A cone truncated by an inclined plane
A cone with a region including its apex cut off by a plane is called atruncated cone; if thetruncation plane is parallel to the cone's base, it is called afrustum.[1] Anelliptical cone is a cone with anelliptical base.[1] Ageneralized cone is the surface created by the set of lines passing through a vertex and every point on a boundary (seeVisual hull).
Proof without words that the volume of a cone is a third of a cylinder of equal diameter and height
1.
A cone and a cylinder haveradiusr andheighth.
2.
The volume ratio is maintained when the height is scaled toh'=r √π.
3.
Decompose it into thin slices.
4.
Using Cavalieri's principle, reshape each slice into a square of the same area.
5.
The pyramid is replicated twice.
6.
Combining them into a cube shows that the volume ratio is 1:3.
Thevolume of any conic solid is one third of the product of the area of the base and the height[4]
In modern mathematics, this formula can easily be computed using calculus — if, where is a coefficient, the integral
Without using calculus, the formula can be proven by comparing the cone to a pyramid and applyingCavalieri'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 themethod of exhaustion. This is essentially the content ofHilbert's third problem – more precisely, not all polyhedral pyramids arescissors 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.[5]
Thecenter 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.
Theslant height of a right circular cone is the distance from any point on thecircle of its base to the apex via a line segment along the surface of the cone. It is given by, where is theradius of the base and is the height. This can be proved by thePythagorean theorem.
Thelateral 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.[4] 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.
Total surface area of a right circular cone, given radius 𝑟 and slant height ℓ
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.
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.
Inimplicit 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 implicitvector equation where
Inprojective geometry, acylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
Inprojective geometry, acylinder is simply a cone whose apex is at infinity.[7] 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 asarctan, in the limit forming aright angle. This is useful in the definition ofdegenerate conics, which require considering thecylindrical conics.
According toG. B. Halsted, a cone is generated similarly to aSteiner conic only with a projectivity andaxial 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'."[8]
The definition of a cone may be extended to higher dimensions; seeconvex cone. In this case, one says that aconvex setC in therealvector space is a cone (with apex at the origin) if for every vectorx inC and every nonnegative real numbera, the vectorax is inC.[2] In this context, the analogues of circular cones are not usually special; in fact one is often interested inpolyhedral cones.
An even more general concept is thetopological cone, which is defined in arbitrary topological spaces.
