arc length of parabola

The parabolaMathworldPlanetmathPlanetmath is one of the quite few plane curvesMathworldPlanetmath, the arc lengthMathworldPlanetmath of which is expressible in closed form; other ones are line, circle (, semicubical parabolaMathworldPlanetmath, logarithmic curve (, catenaryMathworldPlanetmath, tractrixMathworldPlanetmath, cycloidMathworldPlanetmath, clothoidMathworldPlanetmath, astroid, Nielsen’s spiral, logarithmic spiralMathworldPlanetmath.  Determining the arc length of ellipseMathworldPlanetmath ( and hyperbolaMathworldPlanetmath leads to elliptic integralsMathworldPlanetmath.

We evaluate the of the parabola

y=ax2  (a>0) (1)

from the apex (the origin) to the point  (x,ax2).

The usual arc length


where one has made the substitution ( 2ax=:t.  Then one can utilise the result in the entry integration of x2+1 (, whence

s=14a(2ax4a2x2+1+arsinh2ax). (2)

This expression for the parabola arc length becomes especially when the arc is extended from the apex to the end pointPlanetmathPlanetmath(12a,14a)  of the parametre, i.e. the latus rectum; this arc length is


Here,  2+ln(1+2)=:P  is called the universal parabolic constant, since it is common to all parabolas; it is the ratio of the arc to the semiparametre.  This constant appears also for example in the areas of some surfaces of revolutionMathworldPlanetmath (see and Sondow).

Title arc length of parabola
Canonical name ArcLengthOfParabola
Date of creation 2013-03-22 18:57:19
Last modified on 2013-03-22 18:57:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Example
Classification msc 53A04
Classification msc 26A42
Classification msc 26A09
Classification msc 26A06
Synonym closed-form arc lengths
Related topic FamousCurvesInThePlane
Related topic AreaFunctions
Defines universal parabolic constant