Annulus

This page is about mathematics. For other meanings, see also anus and Wiktionary:anus.

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:

<math> r < |z-a| < R. <math>

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 <math>S^1 \times (0,1)<math>.

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

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

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).