楕円の周長

2026-01-26

Takayuki HOSODA

楕円の周長の計算

計算は、第二種完全楕円積分と算術幾何平均（AGM, Arithmetic Geometric Mean）の関係を用いて行われます。
また、ラマヌジャン (Ramanujan) の楕円周近似の値も計算されます。

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).

楕円の周長

楕円の周長の計算

Formulas Used

APPENDIX — Ramanujan's Perimeter of an Ellipse

REFERENCE

SEE ALSO