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29-610: (Redirected from Tapers ) [REDACTED] Look up taper in Wiktionary, the free dictionary. Taper may refer to: Part of an object in the shape of a cone (conical) Taper (transmission line) , a transmission line gradually increasing or decreasing in size Fishing rod taper , a measure of the flexibility of a fishing rod Conically tapered joints, made of ground glass , commonly used in chemistry labs to mate two glassware components fitted with glass tubings Luer taper ,

58-431: A polygonal base is called a pyramid . Depending on the context, "cone" may also mean specifically a convex cone or a projective cone . Cones can also be generalized to higher dimensions . 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

87-420: A sphere is the surface of a ball , for other solid figures it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder . spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)]] Various techniques and tools are used in solid geometry. Among them, analytic geometry and vector techniques have

116-594: A boundary (also see visual hull ). The volume V {\displaystyle V} of any conic solid is one third of the product of the area of the base A B {\displaystyle A_{B}} and the height h {\displaystyle h} In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral ∫ x 2 d x = 1 3 x 3 {\displaystyle \int x^{2}\,dx={\tfrac {1}{3}}x^{3}} Without using calculus,

145-485: A defunct professional basketball team Taper (cymbal) , the reduction in thickness of a cymbal from center to rim Taper pin , used in manufacturing Taper insertion pin , used in body piercing Taper (concert) , a person who records audio concerts, usually via portable setup Taper, a type of men's haircut (see crew cut ) See also [ edit ] Tapering (disambiguation) Tapper (disambiguation) Tapir (disambiguation) Topics referred to by

174-478: A line segment along the surface of the cone. It is given by r 2 + h 2 {\displaystyle {\sqrt {r^{2}+h^{2}}}} , where r {\displaystyle r} is the radius of the base and h {\displaystyle h} is the height. This can be proved by the Pythagorean theorem . The lateral surface area of a right circular cone

203-415: A plane with the lateral surface is a conic section . In general, however, the base may be any shape and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area , 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

232-648: A standardized fitting system used for making leak-free connections between slightly conical syringe tips and needles Tapered thread , a conical screw thread made of a helicoidal ridge wrapped around a cone Machine taper , in machinery and engineering Mark Taper Forum , a theatre in the Los Angeles Music Center A ratio used in aeronautics (see Chord (aeronautics) ) Type of wing configuration in aeronautics ( Wing_configuration#Chord_variation_along_span ) A thin candle Philadelphia Tapers (also New York Tapers and Washington Tapers),

261-477: Is L S A = π r ℓ {\displaystyle LSA=\pi r\ell } where r {\displaystyle r} is the radius of the circle at the bottom of the cone and ℓ {\displaystyle \ell } is the slant height of the cone. The surface area of the bottom circle of a cone is the same as for any circle, π r 2 {\displaystyle \pi r^{2}} . Thus,

290-400: 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 of a cone is

319-430: 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. Solid geometry Solid geometry or stereometry

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348-417: Is different from Wikidata All article disambiguation pages All disambiguation pages Cone A cone is formed by a set of line segments , half-lines , or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle , any one-dimensional quadratic form in

377-461: Is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface ; for example, a solid ball consists of a sphere and its interior . Solid geometry deals with the measurements of volumes of various solids, including pyramids , prisms (and other polyhedrons ), cubes , cylinders , cones (and truncated cones ). The Pythagoreans dealt with

406-714: Is the "height" along the cone. A right solid circular cone with height h {\displaystyle h} and aperture 2 θ {\displaystyle 2\theta } , whose axis is the z {\displaystyle z} coordinate axis and whose apex is the origin, is described parametrically as where s , t , u {\displaystyle s,t,u} range over [ 0 , θ ) {\displaystyle [0,\theta )} , [ 0 , 2 π ) {\displaystyle [0,2\pi )} , and [ 0 , h ] {\displaystyle [0,h]} , respectively. In implicit form,

435-422: The half-angle of the cone, to distinguish it from the aperture. 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 . 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

464-469: The dot product . 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 x 2 + y 2 = z 2   . {\displaystyle x^{2}+y^{2}=z^{2}\ .} From the fact, that the affine image of a conic section is a conic section of

493-551: The regular solids , but the pyramid, prism, cone and cylinder were not studied until the Platonists . Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius . Basic topics in solid geometry and stereometry include: Advanced topics include: Whereas

522-439: The advent of calculus, with the ancient Greeks using the method of exhaustion . This is essentially the content of Hilbert's third problem – 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. The center of mass of a conic solid of uniform density lies one-quarter of

551-432: 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

580-417: 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 . This is useful in the definition of degenerate conics , which require considering the cylindrical conics . According to G. B. Halsted , a cone is generated similarly to a Steiner conic only with a projectivity and axial pencils (not in perspective) rather than

609-403: 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 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

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638-508: 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'." The definition of a cone may be extended to higher dimensions; see convex cone . In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}}

667-586: 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 d {\displaystyle d} , and aperture 2 θ {\displaystyle 2\theta } , is given by the implicit vector equation F ( u ) = 0 {\displaystyle F(u)=0} where where u = ( x , y , z ) {\displaystyle u=(x,y,z)} , and u ⋅ d {\displaystyle u\cdot d} denotes

696-407: The same term [REDACTED] This disambiguation page lists articles associated with the title Taper . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Taper&oldid=1249421828 " Category : Disambiguation pages Hidden categories: Short description

725-411: The same type (ellipse, parabola,...), one gets: 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 projective geometry , a cylinder is simply a cone whose apex is at infinity. Intuitively, if one keeps the base fixed and takes

754-401: The straight line passing through the apex about which the base (and the whole cone) has a circular symmetry . In common usage in elementary geometry , cones are assumed to be right circular , where circular means that the base is a circle 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

783-407: The term "directrix" and the directrix of a conic section, see Dandelin spheres .) The "base radius" of a circular cone is the radius of its base; often this is simply called the radius of the cone. 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 2 θ . In optics , the angle θ is called

812-469: The total surface area of a right circular cone can be expressed as each of the following: The circular sector is obtained by unfolding the surface of one nappe of the cone: The surface of a cone can be parameterized as where θ ∈ [ 0 , 2 π ) {\displaystyle \theta \in [0,2\pi )} is the angle "around" the cone, and h ∈ R {\displaystyle h\in \mathbb {R} }

841-409: The way from the center of the base to the vertex, on the straight line joining the two. For a circular cone with radius r and height h , the base is a circle of area π r 2 {\displaystyle \pi r^{2}} and so the formula for volume becomes The slant height of a right circular cone is the distance from any point on the circle of its base to the apex via

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