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150px, An annulus In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became the dominant language ...
word ''anulus'' or ''annulus'' meaning 'little ring'. The adjectival form is annular (as in
annular eclipse A solar eclipse occurs when a portion of the Earth is engulfed in a shadow cast by the Moon which fully or partially blocks sunlight. This occurs when the Sun, Moon and Earth are aligned. Such alignment coincides with a new moon (syzygy) indi ...
). The open annulus is
topologically equivalentIn mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy is important in the study of iterated functions and more generally d ...
to both the open
cylinder A cylinder (from Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can havi ...
and the
punctured plane This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundame ...
.


Area

The area of an annulus is the difference in the areas of the larger circle of radius and the smaller one of radius : :A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right). The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord (geometry), chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hypotenuse , and the area of the annulus is given by :A = \pi\left(R^2 - r^2\right) = \pi d^2. The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width and area and then Integral, integrating from to : :A = \int_r^R\!\! 2\pi\rho\, d\rho = \pi\left(R^2 - r^2\right). The area of an annulus sector of angle , with measured in radians, is given by : A = \frac \left(R^2 - r^2\right).


Complex structure

In complex analysis an annulus in the complex plane is an open region defined as : r < , z - a, < R. If is , the region is known as the punctured disk (a Disk (mathematics), disk with a Point (mathematics), point hole in the center) of radius around the point . 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 . Each annulus can be holomorphic function, holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map :z \mapsto \frac. The inner radius is then . The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.


See also

* * * * * * , for an alternative approach to the area of the annulus


References


External links


Annulus definition and properties
With interactive animation

With interactive animation {{Compact topological surfaces Circles Elementary geometry Geometric shapes