Ellipse Calculator

An ellipse has a semi-major axis a and a semi-minor axis b, with the focal distance c = √(a² − b²) and the eccentricity e = c ÷ a — 0 for a circle, approaching 1 for a long thin ellipse. The two foci sit at (±c, 0), and every point on the curve has the same total distance 2a to the pair of them, which is exactly what the two-pins-and-a-string construction exploits. The semi-latus rectum b² ÷ a is the half-height directly above a focus, the quantity orbital mechanics uses; the directrix sits at x = ±a ÷ e. Give the page any one of these with a and it recovers b.

The area is πab exactly — the circle's πr² stretched. The perimeter has no closed form in elementary functions: it is 4a·E(e²), the complete elliptic integral of the second kind, which this page evaluates by the arithmetic-geometric mean to about fifteen digits. Ramanujan's second approximation, π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))) with h = ((a − b) ÷ (a + b))², is shown beside it: it is accurate to better than one part in ten million for ordinary shapes, which is why it is the one most calculators quietly use.

The point "at 45°" is ambiguous on an ellipse. The geometric angle is the one you would measure with a protractor from the centre, giving the point at radius r(θ) = ab ÷ √((b cos θ)² + (a sin θ)²). The parametric angle t is the one in the standard parameterisation (a cos t, b sin t), which is the angle on the circumscribed circle before the squash — not the same number except at 0°, 90°, 180° and 270°. Both are given, because mixing them up is one of the commonest errors in CAD, plotting and orbital work. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.