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[[Image:Annulus_area.svg|right|150px|An annulus]]
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In [[mathematics]], an '''annulus''' (the [[Latin]] word for "little ring", with plural ''annuli'') is a ring-shaped object, especially a region bounded by two concentric circles. The adjectival form is '''annular''' (as in [[annular eclipse]]).


The open annulus is [[Homeomorphism|topologically equivalent]] to both the open [[cylinder (geometry)|cylinder]] {{math|''S''<sup>1</sup> &times; (0,1)}} and the [[punctured plane]].
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The area of an annulus is the difference in the areas of the larger [[circle]] of radius {{math|''R''}} and the smaller one of radius {{math|''r''}}:
:<math>A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)\,.</math>
 
The area of an annulus can be obtained from the length of the longest interval that can lie completely inside the annulus, 2*''d'' in the accompanying diagram. This can be proven by the [[Pythagorean theorem]]; the length of the longest interval that can lie completely inside the annulus will be [[tangent]] to the smaller circle and form a right angle with its radius at that point. Therefore ''d'' and ''r'' are the sides of a right angled triangle with hypotenuse ''R'' and the area is given by:
:<math>A = \pi (R^2-r^2) = \pi d^2 \,.</math>
 
The area can also be obtained via [[calculus]] by dividing the annulus up into an infinite number of annuli of [[infinitesimal]] width {{math|''d&rho;''}} and area {{math|2''&pi;&rho; d&rho;''}} ( {{math|= circumference &times; width}}) and then [[integral|integrating]] from ''&rho;'' = ''r'' to ''&rho;'' = ''R'':
:<math>A = \int_r^R 2\pi\rho\, d\rho = \pi(R^2-r^2).</math>
 
The area of an annulus sector of angle {{math|''&theta;''}}, with {{math|''&theta;''}} measured in radians, is given by:
:<math> A = \frac{\theta}{2} (R^2 - r^2) </math>
 
==Complex structure==
In [[complex analysis]] an '''annulus''' {{math|ann''(a; r, R)''}} in the [[complex plane]] is an [[open region]] defined by:
 
:<math> r < |z-a| < R.\,</math>
 
If {{math|''r''}} is {{math|0}}, the region is known as the '''punctured disk''' of radius {{math|''R''}} around the point {{math|''a''}}.
 
As a subset of the complex [[Plane (mathematics)|plane]], an annulus can be considered as a [[Riemann surface]]. The complex structure of an annulus depends only on the ratio {{math|''r''/''R''}}. Each annulus {{math|ann''(a; r, R)''}} can be [[holomorphic function|holomorphically]] mapped to a standard one centered at the origin and with outer radius {{math|1}} by the map
:<math>z \mapsto \frac{z-a}{R}.</math>
The inner radius is then {{math|''r''/''R'' < 1}}.
 
The [[Hadamard three-circle theorem]] is a statement about the maximum value a holomorphic function may take inside an annulus.
 
== See also ==
* [[Annulus theorem]] (or conjecture)
* [[Spherical shell]]
* [[Torus]]
* [[List of geometric shapes]]
 
== External links ==
*[http://www.mathopenref.com/annulus.html Annulus definition and properties] With interactive animation
*[http://www.mathopenref.com/annulusarea.html Area of an annulus, formula] With interactive animation
 
[[Category:Elementary geometry]]
[[Category:Geometric shapes]]

Latest revision as of 17:20, 19 March 2014

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