gives the length of the one-dimensional region reg.


gives the length of the parametrized curve whose Cartesian coordinates xi are functions of t.


interprets the xi as coordinates in the specified coordinate chart.

Details and Options


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Basic Examples  (3)

The length of the line connecting the points , , and :

The length of a circle with radius :

Circumference of a parameterized unit circle:

Length of one revolution of the helix , , expressed in cylindrical coordinates:

Scope  (16)

Special Regions  (3)


Lines can be used in any number of dimensions:

Only a 1D Simplex has meaningful arc length:

It can be embedded in any dimension:


Formula Regions  (2)

The arc length of a circle represented as an ImplicitRegion:

An ellipse:

The arc length of a circle represented as a ParametricRegion:

Using a rational parameterization of the circle:

Mesh Regions  (2)

The arc length of a MeshRegion:

In 3D:

The arc length of a BoundaryMeshRegion in 1D:

Derived Regions  (4)

The portion of a circle intersecting a disk:

The arc length of a Circle intersected with a Triangle:

The arc length of a TransformedRegion:

The measure of a RegionBoundary:

Parametric Formulas  (5)

An infinite curve in polar coordinates with finite length:

The length of the parabola between and :

Arc length specifying metric, coordinate system, and parameters:

Arc length of a curve in higher-dimensional Euclidean space:

The length of a meridian on the two-sphere expressed in stereographic coordinates:

Options  (3)

Assumptions  (1)

The length of a cardioid with arbitrary parameter a:

Adding an assumption that a is positive simplifies the answer:

WorkingPrecision  (2)

Compute the ArcLength using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the ArcLength using 30 digits of precision:

Applications  (8)

The length of a function curve :


Compute the length of a knot:

Compute the length of Jupiter's orbit in meters:

The length can be computed using the polar representation of an ellipse:

Alternatively, use elliptic coordinates with half focal distance and constant :

Extract lines from a graphic and compute their coordinate length:

Color a Lissajous curve by distance traversed:

Color Viviani's curve on the sphere by the fraction of distance traversed:

Find mean linear charge density along a circular wire:

Compute the perimeter length of a Polygon:

Properties & Relations  (6)

ArcLength is a non-negative quantity:

ArcLength[r] is the same as RegionMeasure[r] for any one-dimensional region:

ArcLength for a parametric form is defined as an integral:

ArcLength[x,t,c] is equivalent to RegionMeasure[x,{t},c]:

For a 1D region, ArcLength is defined as the integral of 1 over that region:

The circumference of a 2D region is the ArcLength of its RegionBoundary:

Possible Issues  (2)

The parametric form or ArcLength computes the length of possibly multiple coverings:

The region version computes the length of the image:

The length of a region of dimension other than one is Undefined:

Wolfram Research (2014), ArcLength, Wolfram Language function, (updated 2019).


Wolfram Research (2014), ArcLength, Wolfram Language function, (updated 2019).


Wolfram Language. 2014. "ArcLength." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019.


Wolfram Language. (2014). ArcLength. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_arclength, author="Wolfram Research", title="{ArcLength}", year="2019", howpublished="\url{}", note=[Accessed: 16-June-2024 ]}


@online{reference.wolfram_2024_arclength, organization={Wolfram Research}, title={ArcLength}, year={2019}, url={}, note=[Accessed: 16-June-2024 ]}