Annulus

An annulus

In mathematics, an annulus (the Latin word for "little ring", with plural annuli, more at Wiktionary:anus) 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).

In complex analysis an annulus ann(a; r, R) in the complex plane is an open region defined by:

[itex] r < |z-a| < R. [itex]

If r is 0, the region is known as the punctured disk of radius R around the point a.

The open annulus is topologically equivalent to the open cylinder [itex]S^1 \times (0,1)[itex].

Area

The area of such a (circle shaped) annulus can be obtained in some sense by dividing it up into an infinite number of annuli of infinitesimal width dρ and area equal to 2 π ρ dρ (length (= circumference) × width), which amounts to the formula

[itex] A(r,R) = \int_r^R 2\,\pi\,\rho\,\mathrm d\rho = \pi\,[ R^2-r^2 ] [itex] ,

i.e. the difference of the two circle's areas (but this construction is in fact a "proof" of the formula of the circle's area, obtained for r = 0).

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