# Annulus (mathematics)

In mathematics, an annulus (the Latin word for "little ring", with plural annuli) is a ring-shaped object, especially a region bounded by two concentric circles. The adjectival form is annular (as in annular eclipse).

The open annulus is topologically equivalent to both the open cylinder S1 × (0,1) and the punctured plane.

The area of an annulus is the difference in the areas of the larger circle of radius R and the smaller one of radius r:

${\displaystyle A=\pi R^{2}-\pi r^{2}=\pi (R^{2}-r^{2})\,.}$

The area of an annulus can be obtained from the length of the longest interval that can lie completely inside the annulus, 2*d in the accompanying diagram. This can be proven by the Pythagorean theorem; the length of the longest interval that can lie completely inside the annulus will be tangent to the smaller circle and form a right angle with its radius at that point. Therefore d and r are the sides of a right angled triangle with hypotenuse R and the area is given by:

${\displaystyle A=\pi (R^{2}-r^{2})=\pi d^{2}\,.}$

The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width and area 2πρ dρ and then integrating from ρ = r to ρ = R:

${\displaystyle A=\int _{r}^{R}2\pi \rho \,d\rho =\pi (R^{2}-r^{2}).}$

The area of an annulus sector of angle θ, with θ measured in radians, is given by:

${\displaystyle A={\frac {\theta }{2}}(R^{2}-r^{2})}$

## Complex structure

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

${\displaystyle r<|z-a|

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

${\displaystyle z\mapsto {\frac {z-a}{R}}.}$

The inner radius is then r/R < 1.

The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.