Ellipse

Area & Perimeter Calculator

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Major Axis Length Units
Minor Axis Length Units
AnyCoordinate(x)along Major axis Units
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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 )