# Ellipse Area & Perimeter Calculator

 Unit Number Required Data Entry Major Axis Length Units Minor Axis Length Units AnyCoordinate(x)along Major axis Units AnyCoordinate(y)along Minor axis Units Find Results Point outside/inside Ellipse Square Units Ellipse Area Square Units Ellipse Eccentricity Latus Rectum Length Focus to Focus Length Radius of Auxiliary Circle Radius of Director Circle Length of each EquiConjugate Diameter Perimeter:Ramanujan Formula Perimeter:Ramanujan Revised Formula Perimeter:Hudson Formula Perimeter:Euler Formula Perimeter:Kepler Formula Perimeter:Sipos Formula Perimeter:Naive Formula Perimeter:Peano Formula Perimeter:Maclaurin Formula
a=semi major axis;

b=semi minor axis;

e=[1-(b.b)/(a.a)]^1/2 ; e is eccentricity.

Area = PI*a*b ;

Perimeter Formulae :

Ramanujan Formula- P =~ PI*[3(a+b)-((3a+b)(a+3b))^1/2] ;

Ramanujan Revised Formula* :- P =PI*(a+b)*(1 +3h/(10 +(4-3h)^1/2))

Hudson Formula : - P = PI*(a+b)(64-3h*h)/(64-16h) ; where h=(a-b)^2/(a+b)^2 ;

Euler Formula : - P =2PI*[(a*a+b*b)/2]^1/2 ;

Kepler's Formula :- P =2PI*(ab)^1/2 ;

Sipos Formula:- P =2PI*(a+b)^2 /(a^1/2 + b^1/2)^2 ;

Naive Formula:- P =PI*(a + b) ;

Peano Formula :- P =PI*[3(a+b)/2 - (ab)^1/2 ] ;

Colin Maclaurin Formula ;- P/2PI*a = 1-(1/2)^2 e^2 -(1.3/2.4)^2 *(e^4)/3 -(1.3.5/2.4.6)^3 * (e^6) /5 - ...... infinite convergent series.

The ellipse perimeter is about solving the integral [(a sin t)^2 + (b cos t)^2]^1/2 dt over t=0 to t=2PI which due to symmetry becomes 4 times the value over t=0 to t=PI/2;

or a(1 - (e*cos t)^2)^1/2 integrated over t=0 to 2PI yields the value.e=eccentricity
Let

Integral of a(1 - (e*cos t)^2)^1/2 = I

This is equal to infinite series

1-(1/2)^2 e^2 -(1.3/2.4)^2 *(e^4)/3 -(1.3.5/2.4.6)^3 *(e^6) /5 - ..

whose refined value as per Ramanujan is

Latus Rectum = 2b.b/a
Focus to Focus Length = 2ae
Radius of Auxiliary Circle = a
Radius of Director Circle = Sqrt ( a.a + b.b )
Length of each Equi Conjugate circle =sqrt( (a.a +b.b )/2 )