How this calculator works
Ellipse arc length involves an elliptic integral. This tool integrates one quadrant with adaptive Simpson subdivision, then multiplies by four. The dimensionless integration target is about 10^-12; it is a numerical approximation, not a certified error bound. The table compares both Ramanujan formulas against that numerical reference. Dividing lengths by a keeps the table independent of the selected unit. Use half-axis lengths, not full width and height.
Formula
P = 4a ∫₀^(π/2) sqrt(sin²t + (b/a)² cos²t) dt. Ramanujan I: π[3(a+b) - sqrt((3a+b)(a+3b))]. Ramanujan II: π(a+b)[1 + 3h/(10 + sqrt(4-3h))], h = ((a-b)/(a+b))².
Worked example
For a = 5 m and b = 3 m, numerical circumference is about 25.52699886 m. Ramanujan I gives about 25.52698639 m. With a = b = 2 m, the ellipse is a circle and all methods give 4π m.
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Frequently asked questions
Why compare two approximations?
They have different errors. The difference is easier to see for a very elongated ellipse, so the table reports a relative percentage.
Does swapping the axes change the perimeter?
No. Label the longer semi-axis a and the shorter b before entering them.
Can I start from area?
Yes. Since A = πab, area and either semi-axis determine the other. Measurements that reverse the stated major/minor ordering are rejected.