Annulus (mathematics)

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. Or, it is the area between two concentric circles. The adjectival form is annular (for example, an 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:

\( A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)\,. \)

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:

\( A = \pi (R^2-r^2) = \pi d^2 \,. \)

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

\( A = \int_r^R 2\pi\rho\, d\rho = \pi(R^2-r^2). \)

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

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

Complex structure

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

\( r < |z-a| < R.\, \)

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

\( z \mapsto \frac{z-a}{R}. \)

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.
See also

Annulus theorem (or conjecture)
Spherical shell
Torus

External links

Annulus definition and properties With interactive animation

Mathematics Encyclopedia