# Perimeter

Perimeter[reg]

gives the perimeter of the two-dimensional region reg.

Perimeter[{x1,x2},{s,smin,smax},{t,tmin,tmax}]

gives the perimeter of the parametrized region whose Cartesian coordinates xi are functions of s and t.

Perimeter[{x1,x2},{s,smin,smax},{t,tmin,tmax},chart]

interprets the xi as coordinates in the specified coordinate chart.

# Details and Options

• Perimeter is also known as circumference.
• Coordinate charts in the fourth argument of Perimeter can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
• The following options can be given:
•  AccuracyGoal Infinity digits of absolute accurary sought Assumptions \$Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate conditions on parameters PerformanceGoal \$PerformanceGoal aspects of performance to try to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations
• Perimeter can be used with symbolic regions in GeometricScene.

# Examples

open allclose all

## Basic Examples(4)

The perimeter of a disk:

The perimeter of a 3-4-5 triangle:

The perimeter of an annulus with inner radius 1 and outer radius 2:

The perimeter of a sector expressed in polar coordinates:

## Scope(14)

### Special Regions(4)

The perimeter of a Polygon:

Disk:

Disk can be used as an ellipse:

### Formula Regions(4)

The perimeter of a disk represented as an ImplicitRegion:

The perimeter of a disk represented as a ParametricRegion:

Using a rational parametrization of a disk:

The perimeter of an ImplicitRegion:

The perimeter of a ParametricRegion:

### Mesh Regions(3)

The perimeter of a BoundaryMeshRegion:

The perimeter of a MeshRegion:

The perimeter of a MeshRegion with mixed dimensions:

### Derived Regions(2)

The perimeter of a RegionIntersection:

The perimeter of a TransformedRegion:

### Parametric Formulas(1)

The perimeter of an ellipse with semimajor axes 2 and 1:

The same ellipse in elliptic coordinates:

## Options(2)

### Assumptions(1)

The perimeter assuming the region represents an ellipse:

### WorkingPrecision(1)

Compute the Perimeter using machine arithmetic:

In some cases, the exact answer cannot be computed:

## Applications(4)

A farmer has a 40.5-acre plot of land in the shape of a regular pentagon. How much fence is needed to enclose the plot?

Model the plot as a RegularPolygon of unknown radius:

Determine the radius given the area:

The length of fence needed:

Find the perimeter of a MengerMesh in [0,1]×[0,1]:

Find the general formula for the perimeter of a MengerMesh of order n:

The perimeter tends to infinity even though the regions stay bounded inside [0,1]×[0,1]:

Compute the orbital circumference of Mars:

Verify the result:

The coastline paradox states a country's border is fractal in nature and hence its perimeter is unbounded [more info].

Obtain the polygon representing the United Kingdom:

The perimeter of the United Kingdom with various amounts of sampling points along its border:

## Properties & Relations(3)

For regions with only full-dimensional components, the perimeter is the ArcLength of its boundary:

Regions with nonzero perimeter will have nonzero Area:

Find the perimeter of lamina entities:

Use EntityValue to find the perimeter of a salinon with outer radius 5 and inner radius 1:

Find the perimeter through the region's implicit representation:

## Possible Issues(1)

The perimeter of a region of dimension zero or one is undefined:

The perimeter of a region of dimension three or higher is undefined:

## Neat Examples(3)

Create a gallery of perimeters of special regions:

The perimeter of an implicitly described smiley:

The perimeter of a self-intersecting polygon:

Wolfram Research (2017), Perimeter, Wolfram Language function, https://reference.wolfram.com/language/ref/Perimeter.html (updated 2019).

#### Text

Wolfram Research (2017), Perimeter, Wolfram Language function, https://reference.wolfram.com/language/ref/Perimeter.html (updated 2019).

#### CMS

Wolfram Language. 2017. "Perimeter." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Perimeter.html.

#### APA

Wolfram Language. (2017). Perimeter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Perimeter.html

#### BibTeX

@misc{reference.wolfram_2024_perimeter, author="Wolfram Research", title="{Perimeter}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Perimeter.html}", note=[Accessed: 25-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_perimeter, organization={Wolfram Research}, title={Perimeter}, year={2019}, url={https://reference.wolfram.com/language/ref/Perimeter.html}, note=[Accessed: 25-April-2024 ]}