Perimeter of an ellipse

2026-01-26

Takayuki HOSODA

Perimeter of an ellips calculator

The calculation is performed using the relationship between Complete Elliptic Integrals of the Second Kind and the AGM (Arithmetic Geometric Mean).
The approximation of Ramanujan's elliptic perimeter is also calculated.

Perimeter of an Ellipse
Major axes `a` :
Minor axes `b` :
Perimeter `p` :
Ramanujan's approximation `p_a` :

Calculation results:

Some reference values of accurate (to 65 decimal digits) perimeters by WolframAlpha.

a =  1, b = 1 :  6.2831853071795864769252867665590057683943387987502116419498891846... = 2π (exact)
a =  2, b = 1 :  9.6884482205476761984285031963918294119539183978866008250831163524...
a =  5, b = 1 : 21.010044539689000944699164588473738912894812339134152623096835657...
a = 10, b = 1 : 40.639741801008957425577931011816563791313052134504059403405927819...

Formulas Used

equations: complete elliptic integral of the second kind

Note. m denotes the parameter m = k², with k being the elliptic modulus.
K(m) is the complete elliptic integral of the first kind.

APPENDIX — Ramanujan's Perimeter of an Ellipse

Equations: Ramanujans perimeter of an ellipse

REFERENCE

Complete Elliptic Integral of the First Kind — Wolfram MathWorld
Complete Elliptic Integral of the Second Kind — Wolfram MathWorld
Arithmetic-Geometric Mean — Wolfram MathWorld
Mark B. Villarino, "Ramanujan's Perimeter of an Ellipse"
Elliptic integral — Wikipedia
M. Urabe, An iteration method for calculating the complete elliptic integrals, Numerische Mathematik, Vol. 15, pp. 381–394 (1970).

Perimeter of an ellipse

Perimeter of an ellips calculator

Formulas Used

APPENDIX — Ramanujan's Perimeter of an Ellipse

REFERENCE

SEE ALSO