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 of 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 formula_1 and the minor axis (the shortest diameter) of length formula_2, where the quantities formula_3 and formula_4 are the lengths of the semi-major and semi-minor axes respectively. The exact perimeter formula_5 of an ellipse is given by the integral
formula_6
where formula_7 is the eccentricity of the ellipse, defined as
formula_8
If we define the function
formula_9 
known as the complete elliptic integral of the second kind, the perimeter can be expressed in terms of that function as simply
formula_10
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 a infinite series, is
formula_11
where formula_7 is the eccentricity of the ellipse.
More rapid convergence may be obtained by expanding in terms of formula_13. 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 formula_14, but it may also be written in terns of the double factorial or integer binomial coefficients:
formula_15
The coefficients are slightly smaller (by a factor of formula_16) than the preceding, but also formula_17 is numerically much smaller than formula_18 except at formula_19 and formula_20. For eccentricities less than 0.5 the error is at the limits of double-precision floating-point after the formula_21 term.
Approximations.
Because the exact computation involves elliptic integrals, several approximations have been developed over time.
Ramanujan's approximations.
Indian mathematician Srinivasa Ramanujan proposed multiple approximations.
First approximation.
formula_22
Second approximation.
formula_23
where formula_24.
Final approximation.
The final approximation in Ramanujan's notes on the perimeter of the ellipse is regarded as one of his most mysterious equations. It is
formula_25
where 
formula_26
and formula_7 is the eccentricity of the ellipse.
Ramanujan did not provide any rationale for this formula.
Simple arithmetic-geometric mean approximation.
formula_28
This formula is simpler than most perimeter formulas but not very accurate, and entirely unsuitable for eccentric ellipses.
Approximations made from programs.
In an online video about the perimeter of an ellipse, recreational mathematician and YouTuber Matt Parker, using a computer program, calculated numerous approximations for the perimeter of an ellipse. One approximation Parker found (worse for most eccentricities than any of Ramanujan's approximations) was
formula_29