Ellipse Perimeter Calculator

About this calculator

Unlike a circle's circumference, an ellipse's perimeter has no simple closed-form formula — it requires an elliptic integral. This calculator uses Ramanujan's approximation, P ≈ π[3(a+b) − √((3a+b)(a+3b))], which is accurate to within a tiny fraction of a percent for any ratio of semi-major axis a to semi-minor axis b, far better than the naive average-radius estimate. It also computes the area, A = π·a·b, which does have an exact formula.

Elliptical shapes appear in orbital mechanics (planetary and satellite orbits are ellipses per Kepler's first law), engineering design of elliptical gears, tanks, and arches, and architecture and landscaping for elliptical gardens, tracks, and domes.

Engineers and architects laying out elliptical structures, students studying Kepler's laws or conic sections, and CAD/CNC users who need the exact perimeter to cut material efficiently all use this calculator, entering just the two semi-axis lengths.