# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# 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 OptionsDetails and Options

• 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,tmin,tmax}], 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:
•  Assumptions \$Assumptions assumptions to make about parameters Method Automatic method to use WorkingPrecision Infinity the precision used in internal computations
• Specific methods include:
•  Automatic automatic method selection "Integrate" exact symbolic integration "NIntegrate" numeric integration
• Additional method suboptions can be given in the form Method->{,opts}.
• Any option of Integrate or NIntegrate can be passed as a method suboption to the corresponding method.
• 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.

## ExamplesExamplesopen allclose all

### Basic Examples  (4)Basic Examples  (4)

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

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The length of a circle with radius :

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Circumference of a parameterized unit circle:

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

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The length of a region of dimension zero is 0:

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The length of a region of dimension two or higher is :

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