Ellipse Area
A = π·a·bP ≈ π·(a+b)·(1+3h/(10+√(4-3h))) [Ramanujan]The area of an ellipse is π multiplied by the two semi-axes: the major and the minor. It is the direct generalisation of the circle formula, where both semi-axes equal the radius.
A semi-axis is half the axis — the distance from the centre to the edge in that direction. Confusing axis with semi-axis quadruples the result, the same mistake as diameter versus radius on a circle.
Unlike the area, an ellipse’s perimeter cannot be expressed in closed form with elementary functions. It involves an elliptic integral, which has no solution in simple terms.
Every calculator therefore uses an approximation. This one uses Ramanujan’s, published in 1914, which is remarkably accurate: the error stays below one part in ten million for ellipses of moderate eccentricity.
It is a rare case in elementary geometry where the exact answer simply does not exist in closed form — worth knowing when comparing results across tools, which may use different approximations.
- Planetary orbits, which are elliptical with the central body at one focus.
- Cross-sections of flattened pipework and oval ducts.
- Oval tables, mirrors and worktops in joinery.
- Running tracks and oval beds in landscaping.
Frequently asked questions
The semi-axis — the distance from the centre to the edge. Using the full axis in both fields multiplies the area by four.
Because an ellipse’s perimeter has no closed form with elementary functions. Every calculator approximates; this one uses Ramanujan’s formula, with negligible practical error.
Yes. The circle is the special case where both semi-axes coincide, and the area formula reduces to π times the radius squared.
For the area, results should match. For the perimeter, small differences are expected because each tool may use a different approximation.