gives the perimeter of the two-dimensional region reg.


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


interprets the xi as coordinates in the specified coordinate chart.

Details and Options


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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 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:

Introduced in 2017
Updated in 2019