Ellipse Perimeter Calculator

Formulas

Exact: P = 4 a E(e), with e = \(\sqrt{{1 - b^2/a^2}}\) and E the complete elliptic integral of the second kind.

• Ramanujan I: P \approx \pi [3(a+b) - \sqrt{{(3a+b)(a+3b)}}]
• Ramanujan II: h = ((a-b)^2)/(a+b)^2, P \approx \pi(a+b)[1 + 3h/(10 + \sqrt{{4 - 3h}})]
• Kepler: P \approx \pi\,\sqrt{{2(a^2 + b^2)}}

What is Ellipse Perimeter?

The perimeter of an ellipse, sometimes called its circumference, is the total distance around the oval defined by semi‑major axis a and semi‑minor axis b. Unlike a circle, there is no elementary closed form for this length. The exact value uses the complete elliptic integral of the second kind: P = 4 a E(e), where e = sqrt(1 − b²/a²) is the eccentricity. Because elliptic integrals are special functions, practitioners often use approximations. Ramanujan’s first formula is P ≈ π [3(a + b) − √((3a + b)(a + 3b))]. His second formula refines accuracy using h = ((a − b)²)/(a + b)² with P ≈ π(a + b) [1 + 3h/(10 + √(4 − 3h))]. Both give results within tiny fractions of a percent for most shapes. For highly eccentric ellipses, numerical evaluation of E(e) provides a reference. In applications, a represents the longer semi axis and b the shorter, measured in consistent units. Perimeter scales linearly with units, so converting inputs converts perimeter accordingly. Understanding ellipse perimeter supports design of racetracks, gears, orbits, optical apertures, and any geometry involving ovals. This calculator presents exact and approximate methods, explains formulas, and quantifies error so you can choose the best approach.