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Ellipse Perimeter (Ramanujan Approximation)

Also known as perimeter of an ellipse · circumference of an ellipse · oval circumference

P≈π[3(a+b)−(3a+b)(a+3b)]P \approx \pi \left[ 3(a+b) - \sqrt{(3a+b)(a+3b)} \right]

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Constant used — built into this formula, no need to enter
π=3.141592653589793\pi = 3.141592653589793Pi · exact

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Here is one of mathematics' famous asymmetries: an ellipse's area is the effortless πab\pi ab, but its perimeter has no elementary closed form at all. The exact answer is a complete elliptic integral of the second kind, which cannot be written with ordinary functions — so every practical formula is an approximation.

This one is Srinivasa Ramanujan's second approximation, published in 1914, and it is astonishingly good: for ellipses that are not extremely elongated its error is smaller than one part in a billion, far beyond any measurement you could take of a real object. Set a=ba = b and it returns 2πa2\pi a exactly, the circle's circumference, as any honest approximation must.

The problem is not academic. Elliptical duct, oval tanks, running-track lanes and planetary orbits all need a perimeter, and Ramanujan's expression is what engineering handbooks quietly use.

Ellipse Perimeter (Ramanujan Approximation)
P≈π[3(a+b)−(3a+b)(a+3b)]P \approx \pi \left[ 3(a+b) - \sqrt{(3a+b)(a+3b)} \right]
Where
  • PP= Perimeter
  • aa= Semi-major axis
  • bb= Semi-minor axis
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