In mathematics, 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 a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

word ''anulus'' or ''annulus'' meaning 'little ring'. The adjectival form is annular (as in

annular eclipse A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of the Earth, totally or partially. Such an alignment occurs during an eclipse season, approximately every six mon ...

). The open annulus is

topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fun ...

to both the open

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

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 funda ...

Area

The area of an annulus is the difference in the areas of the larger

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

of radius and the smaller one of radius : :

A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right).

annuli_with_same_area_around_unit_regular_polygons

The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

by dividing the annulus up into an infinite number of annuli of infinitesimal width and area and then 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 In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...

defined as :

r < , z - a,  < R.

If is , the region is known as the punctured disk (a disk with a point hole in the center) of radius around the point . As a subset of the complex

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

, an annulus can be considered as a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...

. The complex structure of an annulus depends only on the ratio . Each annulus can be 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 In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions. Let f(z) be a holomorphic function on the annulus :r_1\leq\left, z\ \leq r_3. Let M(r) be the maximum of , f ...

is a statement about the maximum value a holomorphic function may take inside an annulus.

References

External links

Annulus definition and properties
With interactive animation

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

Area

Complex structure

See also

References

External links