Inmathematics, anannulus (pl.:annuli orannuluses) is the region between two concentric circles. Informally, it is shaped like a ring or ahardware washer. The word "annulus" is borrowed from theLatin wordanulus orannulus meaning 'little ring'. The adjectival form isannular (as inannular eclipse).
The open annulus istopologically equivalent to both the opencylinderS1 × (0,1) and thepunctured plane.
The area of an annulus is the difference in the areas of the largercircle of radiusR and the smaller one of radiusr:
The area of an annulus is determined by the length of the longestline segment within the annulus, which is thechord tangent to the inner circle,2d in the accompanying diagram. That can be shown using thePythagorean theorem since this line istangent to the smaller circle and perpendicular to its radius at that point, sod andr are sides of a right-angled triangle with hypotenuseR, and the area of the annulus is given by
The area can also be obtained viacalculus by dividing the annulus up into an infinite number of annuli ofinfinitesimal widthdρ and area2πρ dρ and thenintegrating fromρ =r toρ =R:
The area of anannulus sector (the region between twocircular sectors with overlapping radii) of angleθ, withθ measured in radians, is given by
Incomplex analysis anannulusann(a;r,R) in thecomplex plane is anopen region defined as
If, the region is known as thepunctured disk (adisk with apoint hole in the center) of radiusR around the pointa.
As a subset of the complexplane, an annulus can be considered as aRiemann surface. The complex structure of an annulus depends only on the ratior/R. Each annulusann(a;r,R) can beholomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map
The inner radius is thenr/R < 1.
TheHadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.
TheJoukowsky transformconformally maps an annulus onto anellipse with a slit cut between foci.
