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Annulus (mathematics)

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	Topics Annulus (mathematics) Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , an annulus (the LatinLatin Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome.... word for "little ring", with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. The adjective form is annular (for example, an annular eclipse). The open annulus is topologically equivalent to both the open cylinderCylinder (geometry) In mathematics, a cylinder is a quadric, i.e.... and the punctured plane. The area of such an annulus is given by the difference in the areas of a circleCircle In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed poi... of radius R and one of radius r: Interestingly, the area of an annulus can also be obtained by multiplying pi by the square of half of the length of the longest interval that can lie completely inside the annulus. This can be proven by the Pythagorean theoremPythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sid... ; the length of the longest interval that can lie completely inside the annulus will be tangentTangent In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonomet... to the smaller circle. Given the above formula for area, half of the length of the interval will actually form a right angle, along with radius r, to form diagonal R. This result can be obtained via calculusCalculus Calculus is a central branch of mathematics, developed from algebra and geometry.... by dividing the annulus up into an infinite number of annuli of infinitesimalInfinitesimal In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positiv... width and area ( = circumference × width) and then integratingIntegral Overview In calculus, the integral of a function is an extension of the concept of a sum.... from to : Complex structure In complex analysisComplex analysis Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use i... an annulus ann(a; r, R) in the complex planeComplex plane In mathematics, the complex plane is a geometric space of the complex numbers as set up by the real axis and the ortho... is an open region defined by: If r is 0, the region is known as the punctured disk of radius R around the point a. As a subset of the complex planeFacts About Plane (mathematics) In mathematics, a plane is a fundamental two-dimensional object.... , an annulus can be considered as a Riemann surfaceRiemann surface In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional comp... . The complex structure of an annulus depends only on the ratio r/R. Each annulus ann(a; r, R) can be holomorphicallyFacts About Holomorphic function Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of t... mapped to a standard one centered at the origin and with outer radius 1 by the map The inner radius is then r/R < 1. The Hadamard three-circle theoremHadamard three-circle theorem In complex analysis, a branch of mathematics, the... is a statement about the maximum value a holomorphic function may take inside an annulus. See also CylinderFacts About Cylinder (geometry) In mathematics, a cylinder is a quadric, i.e.... TorusFacts About Torus GeometryIn geometry, a torus is a doughnut-shaped surface of revolution generated by revolving a circle in three dimensiona... External links With interactive animation With interactive animation All Math Words Encyclopedia for grades 7-10.