Cone (geometry) - Citizendia

This article is about the geometric object. For other uses, see Cone.

A right circular cone and an oblique circular cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined More precisely, it is the solid figure bounded by a plane base and the surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property The perimeter is the distance around a given two-dimensional object 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. Generally 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 is technically a cone with a polygonal base. Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In common usage in elementary geometry, 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 to its plane, and circular means that the base is a circle. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.

1 Other mathematical meanings
2 Further terminology
3 Formula
- 3.1 Right circular cone
4 See also
5 External links

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, and the intersection of a plane with this surface is a conic section. In Geometry, a ( general) conical surface is the unbounded Surface formed by the union of all the straight lines that pass through a fixed In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface 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 or a projective cone. In Linear algebra, a convex cone is a Subset of a Vector space that is closed under Linear combinations with positive coefficients 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

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. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Geometry, a nondegenerate Conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus Each )

The base radius of a circular cone is the radius of its base; often this is simply called the radius of the cone. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers The aperture 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 parallel to its base is called a truncated cone or frustum. Elements special cases and related concepts Each plane section is a base of the frustum An elliptical cone is a cone with an elliptical base. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a

Formula

See also: Cone (geometry) proofs. Volume Claim The volume of a conic solid whose base has area b and whose height is h is {1\over 3} b h.

The volume $V$ of any conic solid is one third the area of the base $b$ times the height $h$ (the perpendicular distance from the base to the apex). The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically

$V = \frac{1}{3} b h$

The center of mass 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

$V = \frac{1}{3} \pi r^2 h.$

For a right circular cone, the surface area $A$ is

$A =\pi r^2 + \pi r s\,$ where $s = \sqrt{r^2 + h^2}$ is the slant height. 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 slant height of a Right circular cone is the distance from any point on the Circle to the apex of the cone

The first term in the area formula, $π r 2$ , is the area of the base, while the second term, $π r s$ , is the area of the lateral surface.

A right circular cone with height $h$ and aperture $2θ$ , whose axis is the $z$ coordinate axis and whose apex is the origin, is described parametrically as

$S(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)$

where $s, t, u$ range over $[0,θ)$ , $[0,2π)$ , and $[0, h]$ , respectively.

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

$\{ S(x,y,z) \leq 0, z\geq 0, z\leq h\}$ ,

where

$S(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.\,$ . In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly"

More generally, a right circular cone with vertex at the origin, axis parallel to the vector $d$ , and aperture $2θ$ , is given by the implicit vector equation $S (u) = 0$ where

$S(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2$ or $S(u) = u \cdot d - |d| |u| \cos \theta$

where $u = (x, y, z)$ , and $u \cdot d$ denotes the dot product. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

External links

In Topology, especially Algebraic topology, the cone CX of a Topological space X is the Quotient space: Democritus ( Greek:) was a pre-Socratic Greek Materialist Philosopher (born at Abdera in Thrace ca Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic In Geometry, a Surface S is ruled if through every point of S there is a straight line that lies on S In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} + 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 Math Is Fun (or Maths Is Fun in British English is an educational website maintained by Rod Pierce devoted to the concept that Mathematics is indeed fun

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Contents

Other mathematical meanings

Further terminology

Formula

Right circular cone

See also

External links