perimeter of astroid


The astroid

x23+y23=a23

can be presented in the parametric form

x=acos3φ,y=asin3φ,

where the polar angle φ gets the values from 0 to 2π. The curve consists of four congruentMathworldPlanetmath arcs, one of which is obtained letting  0φπ2. Thus the whole perimeter of the astroid is

s=40π2(dxdφ)2+(dydφ)2𝑑φ.

This expression is easily simplified to

s=12a0π2sinφcosφdφ

giving the result

s=6a0π2sin2φdφ=-3a/0π2cos2φ=6a.
Title perimeter of astroid
Canonical name PerimeterOfAstroid
Date of creation 2013-03-22 17:14:06
Last modified on 2013-03-22 17:14:06
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 26B15
Synonym perimetre of astroid
Related topic Parameter
Related topic ClairautsEquation
Related topic SubstitutionNotation