Unlike most other elementary shapes, such as the circle and square, there is no closed-form expression for the perimeter of an ellipse. Throughout history, a large number of closed-form approximations and expressions in terms of integrals or series have been given for the perimeter of an ellipse.
Exact value
Elliptic integral
An ellipse is defined by two axes: the major axis (the longest diameter) of length 2 a {\displaystyle 2a} and the minor axis (the shortest diameter) of length 2 b {\displaystyle 2b} , where the quantities a {\displaystyle a} and b {\displaystyle b} are the lengths of the semi-major and semi-minor axes respectively. The exact perimeter P {\displaystyle P} of an ellipse is given by the integral P = 4 a ∫ 0 π / 2 1 − e 2 sin 2 θ d θ , {\displaystyle P=4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta ,} where e {\displaystyle e} is the eccentricity of the ellipse, defined as e = 1 − b 2 a 2 . {\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.} If we define the function E ( x ) = ∫ 0 π / 2 1 − x sin 2 θ d θ , {\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x\sin ^{2}\theta }}\ d\theta ,} known as the complete elliptic integral of the second kind, the perimeter can be expressed in terms of that function simply as P = 4 a E ( e 2 ) . {\displaystyle P=4aE(e^{2}).} The integral used to find the perimeter does not have a closed-form solution in terms of elementary functions.
Infinite sums Another solution for the perimeter, this time using the sum of an infinite series, is P = 2 a π ( 1 − ∑ n = 1 ∞ ( 2 n ! ) 2 ( 2 n ⋅ n ! ) 4 ⋅ e 2 n 2 n − 1 ) , {\displaystyle P=2a\pi \left(1-\sum _{n=1}^{\infty }{\frac {(2n!)^{2}}{(2^{n}\cdot n!)^{4}}}\cdot {\frac {e^{2n}}{2n-1}}\right),} where e {\displaystyle e} is the eccentricity of the ellipse. More rapid convergence may be obtained by expanding in terms of h = ( a − b ) 2 / ( a + b ) 2 {\displaystyle h=(a-b)^{2}/(a+b)^{2}} . Found by James Ivory, Bessel and Kummer, there are several equivalent ways to write it. The most concise is in terms of the binomial coefficient with n = 1 / 2 {\displaystyle n=1/2} , but it may also be written in terms of the double factorial or integer binomial coefficients:
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