How to find the perimeter of an ellipse?

Question

for example a=100,b=62.5

Answer 1

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One of the best approximation( range -0.04%) is obtained from the following formula,
P = pi* (a+b) [ 1 + 3h / ( 10+ (4-3h)^(1/2) ) ]
Where h= (a-b)^2/(a + b)^2
This was Ramanujan's second formula

Lindler's formula has an error range of - 0.598%
it states,
P = pi* (a+b) [ 1 + h/8 ]^ 2

Hudson's formula gives an error range of -0.19%
according to this,

P= pi* (a+b) (64 - 3 h^2)/(64 - 16 h)
A more precise Padé approximant consists of the optimized ratio of two quadratic polynomials of h and leads to the following formula, credited to Jacobsen and Waadeland (1985). This expression gives a relative error for the perimeter of a round ellipse (low e) roughly equal to -33 e^20/2^38 and a relative error of about -0.0888 % for a flat ellipse. This is directly comparable with the accuracy of Ramanujan's second formula , which is 5.5 times better for the circumference of a round ellipse [and about 2.2 times better for a flat ellipse
P =pi (a+b)(256 - 48 h - 21 h^2)/(256 - 112 h + 3 h^2)

There are many other such formulae. Try these links

Source(s):

http://home.att.net/~numericana/answer/e…
http://mathforum.org/dr.math/faq/formula…
Hope that helps.

Answer 2

It's a big equation with an infinite series solution.

Source(s):

http://en.wikipedia.org/wiki/Ellipse#Cir…

Answer 3

If you want it exact (good luck with this!) you will need to solve the following definite integral between x=0 and x=pi/2

4*a*INT(SQRT(1 - [(a^2-b^2)/a]*sin^2x) dx)

If you want a 5-10% approximation do

2*pi*SQRT((a^2+b^2)/2)

I have also seen

PI * (a+b) as an approximation, although depending on the excentriciy the error could reach 20%. If it almost a circle it will be very good.

Answer 4

This will give a close approximation, but not exact.
P = pi√ 2(a²+b²)

Apparently there is no exact formula, but check this site for all the formulae:

http://home.att.net/~numericana/answer/e…

Answer 5

P=2 x Pi x SQRT((a2+b2)/2)

That is 523,9259226 for your a and b given.

That is approximate, because you really need to find the value of an elliptic integral of the second kind to get the exact number.

Answer 6

Easy question; very difficult answer.
You need elliptical integrals.
Please forget it, unless you are a professional.
Then you do not need yahoo! answers.

Th