In mathematics, an annulus (the Latin 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 the open cylinder $S^1\; \backslash times\; (0,1)$.

The area of such an annulus is given by the difference in the areas of a circle of radius R and one of radius r:

$A\; =\; \backslash pi(R^2\; -\; r^2)\backslash ,$

This result can be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width $d\backslash rho$ and area $2\backslash pi\backslash rho\; d\backslash rho$ ( = circumference × width) and then integrating from $\backslash rho\; =\; r$ to $\backslash rho\; =\; R$:

$A\; =\; \backslash int\_r^R\; 2\backslash pi\backslash rho\backslash ,\; d\backslash rho\; =\; \backslash pi(R^2-r^2).$

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.

Complex structure

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In complex analysis an annulus ann(a; r, R) in the complex plane 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 plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio r/R. Each annulus ann(a; r, R) can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map

$z\; \backslash mapsto\; \backslash frac\{z-a\}\{R\}.$

The inner radius is then r/R < 1.

See also

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• Cylinder
• Torus

Elementary geometry | Geometric shapes

Annulus | Kreisring | Corona (matemática) | Corona circolare | Anel (topologia)