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ArcLength
✖
ArcLength

✖
ArcLength[reg]

gives the length of the one-dimensional region reg.

✖
ArcLength[{x1,…,xn},{t,tmin,tmax}]

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

✖
ArcLength[{x1,…,xn},{t,tmin,tmax},chart]

interprets the xi as coordinates in the specified coordinate chart.

Details and Options

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-6b2eir
Out[1]=1

The length of a circle with radius :

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-fh516u
Out[2]=2

Circumference of a parameterized unit circle:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-1m2mrf
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-8oa2el
Out[1]=1

Scope  (16)Survey of the scope of standard use cases

Special Regions  (3)

Line:

In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-i6gtwd
Out[4]=4
In[4]:=4
✖
https://wolfram.com/xid/0cg41t8c6-bze2jo
Out[4]=4

Lines can be used in any number of dimensions:

In[5]:=5
✖
https://wolfram.com/xid/0cg41t8c6-gzu97q
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cg41t8c6-bt45lk
Out[6]=6

Only a 1D Simplex has meaningful arc length:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-l20po6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-dghzzm
Out[2]=2

It can be embedded in any dimension:

In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-dxhzge
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cg41t8c6-fwfbek
Out[4]=4

Circle:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-fsw7u6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-dec5e3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-4zyid
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cg41t8c6-ew23om
Out[4]=4

Formula Regions  (2)

The arc length of a circle represented as an ImplicitRegion:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-lpgmaf
Out[1]=1

An ellipse:

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-ge40ck
Out[2]=2

The arc length of a circle represented as a ParametricRegion:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-cbaozy
Out[1]=1

Using a rational parameterization of the circle:

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-idqj43
Out[2]=2

Mesh Regions  (2)

The arc length of a MeshRegion:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-fqlaq1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-bbivoq
Out[2]=2

In 3D:

In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-0v63v
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cg41t8c6-gfeksk
Out[4]=4

The arc length of a BoundaryMeshRegion in 1D:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-d7iagg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-gh14gg
Out[2]=2

Derived Regions  (4)

The portion of a circle intersecting a disk:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-keesdp
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-bnd08k
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-31793a
Out[3]=3

The arc length of a Circle intersected with a Triangle:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-frmru
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-bogjqw
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-xu1hh
Out[3]=3

The arc length of a TransformedRegion:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-cy0hoy
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-tx9fp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-031qh
Out[3]=3

The measure of a RegionBoundary:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-c1e66s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-tifyf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-fclgc8
Out[3]=3

Parametric Formulas  (5)

An infinite curve in polar coordinates with finite length:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-j9ke5e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-zbc7h9
Out[2]=2

The length of the parabola between and :

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-zqfzd3
Out[1]=1

Arc length specifying metric, coordinate system, and parameters:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-bxhu6q
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-ur3sdt
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-oienkx
Out[1]=1

Options  (3)Common values & functionality for each option

Assumptions  (1)

The length of a cardioid with arbitrary parameter a:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-kwi8kb
Out[1]=1

Adding an assumption that a is positive simplifies the answer:

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-r1y92d
Out[2]=2

WorkingPrecision  (2)

Compute the ArcLength using machine arithmetic:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-syuukw
Out[1]=1

In some cases, the exact answer cannot be computed:

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-2e3szr
Out[2]=2

Find the ArcLength using 30 digits of precision:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-5n8cz9
Out[1]=1

Applications  (8)Sample problems that can be solved with this function

The length of a function curve :

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-3dxxi
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-y3pu1
Out[2]=2

Equivalently:

In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-knm0b5
Out[3]=3

Compute the length of a knot:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-xi7ee0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-b2figk
Out[2]=2

Compute the length of Jupiter's orbit in meters:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-nb5wc7
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-nsxiw8
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-r3p4h0
Out[3]=3

Extract lines from a graphic and compute their coordinate length:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-820khw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-wao2ma
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-r7sxgf
Out[3]=3

Color a Lissajous curve by distance traversed:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-hvq85f
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-pnfqt2
Out[1]=1

Find mean linear charge density along a circular wire:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-zt1i8
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-bl3d
Out[2]=2

Compute the perimeter length of a Polygon:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-gexqnf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-ba8htg
Out[2]=2

Properties & Relations  (6)Properties of the function, and connections to other functions

ArcLength is a non-negative quantity:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-d7kpbu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-dz89bz
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-djfuhh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-brf409
Out[2]=2

ArcLength for a parametric form is defined as an integral:

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-ckdol5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-fi6e9t
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-uuk9as
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-hmf8as
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-uh5ef0
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-g4cfg
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-m1f8bd
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-h5h6wy
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-e2llin
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-f51e7
Out[1]=1

The region version computes the length of the image:

In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-grcil2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg41t8c6-e3v23y
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0cg41t8c6-s1t4gx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg41t8c6-yy6k67
Out[2]=2
Wolfram Research (2014), ArcLength, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcLength.html (updated 2019).
✖
Wolfram Research (2014), ArcLength, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcLength.html (updated 2019).

Text

Wolfram Research (2014), ArcLength, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcLength.html (updated 2019).

✖
Wolfram Research (2014), ArcLength, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcLength.html (updated 2019).

CMS

Wolfram Language. 2014. "ArcLength." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/ArcLength.html.

✖
Wolfram Language. 2014. "ArcLength." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/ArcLength.html.

APA

Wolfram Language. (2014). ArcLength. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcLength.html

✖
Wolfram Language. (2014). ArcLength. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcLength.html

BibTeX

@misc{reference.wolfram_2025_arclength, author="Wolfram Research", title="{ArcLength}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/ArcLength.html}", note=[Accessed: 24-March-2025 ]}

✖
@misc{reference.wolfram_2025_arclength, author="Wolfram Research", title="{ArcLength}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/ArcLength.html}", note=[Accessed: 24-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arclength, organization={Wolfram Research}, title={ArcLength}, year={2019}, url={https://reference.wolfram.com/language/ref/ArcLength.html}, note=[Accessed: 24-March-2025 ]}

✖
@online{reference.wolfram_2025_arclength, organization={Wolfram Research}, title={ArcLength}, year={2019}, url={https://reference.wolfram.com/language/ref/ArcLength.html}, note=[Accessed: 24-March-2025 ]}