# Annulus

Annulus[{x,y},{rinner,router}]

represents an annulus centered at {x,y} with inner radius rinner and outer radius router.

Annulus[{x,y},{rinner,router},{θ1,θ2}]

represents an annulus from angle θ1 to θ2.

# Details • Annulus is also known as a punctured disk.
• Annulus can be used as a geometric region and a graphics primitive.
• Annulus[] is equivalent to Annulus[{0,0},{1/2,1}].
• Annulus[r] is equivalent to Annulus[{0,0},{r/2,r}].
• Annulus[{rinner,router}] is equivalent to Annulus[{0,0},{rinner,router}].
• • Annulus represents the filled region or .
• Annulus allows and .
• Angles are measured in radians counterclockwise from the positive x direction.
• Annulus can be used in Graphics.
• In graphics, the points {xi,yi} can be Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, and color.

# Examples

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## Basic Examples(4)

The standard annulus:

An annulus sector:

Differently styled annuli:

Get the Area of an annulus:

The area of an annulus slice:

## Scope(17)

### Graphics(7)

#### Specification(4)

Specify centers:

An annulus sector:

Short form for a basic annulus at the origin:

#### Styling(2)

Color directives specify the face color of each annulus:

FaceForm and EdgeForm can be used to specify the styles of the interiors and boundaries:

Boundaries of an annulus:

#### Coordinates(1)

Points can be Dynamic:

### Regions(10)

Embedding dimension:

Geometric dimension:

Point membership test:

Get conditions for point membership:

Area:

Centroid:

Distance from a point:

The distance to the nearest point in the basic annulus:

Signed distance from a point:

Signed distance to the basic annulus:

Nearest point in the region:

Nearest points:

An annulus is bounded:

Get its range:

Integrate over an annulus:

Optimize over an annulus:

Solve equations in an annulus:

## Applications(4)

Plot a function over an annulus:

A capacitor is fundamentally a pair of conducting plates with an insulator between them. The capacitor's capacitance is a function of the plates' separation and the area of overlap, which allow for the creation of variable capacitors that are made of a set of half annuli that can be rotated to change the overlap between them. Make a function that returns the arc of overlap for two concentric annuli:

Then make a function that finds the area of overlap:

Knowing that , make a function that calculates the capacitance:

Create an interactive and graphical representation of a variable capacitor that displays its capacitance (in picofarads):

As a tree grows, its cross section gains growth rings of various widths, depending on the year's climate. You can use annuli to create a pattern similar to growth rings. First, generate a list of random but adjacent inner and outer radii:

Create a table of concentric annuli with a random (brownish) color:

Finally, visualize all of the annuli, along with a disk with the radius equal to the inner radius of the smallest annulus for the core:

Under certain conditions, an annular solar eclipse may occur. The term annular comes from annulus, since the relative sizes of the Sun and Moon in the sky are such that part of the Sun (the corona) is visible around the Moon, forming a bright annulus. One can make an interactive demonstration of this phenomenon, and make a bright white annulus appear at the moment of annular eclipse:

## Properties & Relations(6)

A Disk is the limit of Annulus as goes to 0:

A Circle is the limit of Annulus as goes to :

An annulus is the closure of RegionDifference between two concentric Disk regions:

ImplicitRegion can represent any Annulus:

ParametricRegion can represent an Annulus:

Annulus is the points less than from a circle with radius :

## Neat Examples(3)

Random annulus collections:

A family of annuli:

Digital petals: