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ArcLength—Wolfram Language Documentation

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• See Also
  • RegionMeasure
  • Perimeter
  • ArcCurvature
  • FrenetSerretSystem
  • CoordinateChartData
  • Area
  • SurfaceArea
  • Volume
  • Integrate
  • NIntegrate
  • CoordinateTransform
  • GeometricScene
• Related Guides
  • Region Properties and Measures
  • Vector Analysis
  • Solid Geometry
  • Calculus
  • Plane Geometry
  • Precollege Education
  • Synthetic Geometry
  • Interval Arithmetic

ArcLength[reg]

gives the length of the one-dimensional region reg.

ArcLength[{x₁,…,x_n},{t,t_min,t_max}]

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

ArcLength[{x₁,…,x_n},{t,t_min,t_max},chart]

interprets the x_i as coordinates in the specified coordinate chart.

• ArcLength is also known as length or curve length.
• A one-dimensional region can be embedded in any dimension greater than or equal to one.
• The ArcLength of a curve in Cartesian coordinates is .
• In a general coordinate chart, the ArcLength of a parametric curve is given by , where is the metric.
• In ArcLength[x,{t,t_min,t_max}], if x is a scalar, ArcLength returns the length of the parametric curve {t,x}.
• Coordinate charts in the third argument of ArcLength 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:
• Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData.
• ArcLength can be used with symbolic regions in GeometricScene.

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:

Line:

Lines can be used in any number of dimensions:

Only a 1D Simplex has meaningful arc length:

It can be embedded in any dimension:

Circle:

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:

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:

The length of a cardioid with arbitrary parameter a:

Adding an assumption that a is positive simplifies the answer:

Compute the ArcLength using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the ArcLength using 30 digits of precision:

The length of a function curve :

Equivalently:

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:

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:

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