Ellipse Area Calculator
Find the area of an ellipse from its semi-major and semi-minor axes, with an accurate Ramanujan estimate of the circumference.
How it works
Area = π · a · b · Circumference ≈ π · [3(a + b) − √((3a + b)(a + 3b))]
An ellipse is a stretched circle, so its area follows the same pattern as πr² but with the two different half-widths in place of a single radius: π times the semi-major axis a times the semi-minor axis b. When a = b it collapses to a circle of area πr². The circumference has no simple closed form, so this uses Ramanujan’s second approximation, which is accurate to a tiny fraction of a percent for ordinary ellipses.
Worked example
For a = 5 and b = 3: area = π × 5 × 3 = 15π ≈ 47.1239 square units. The circumference ≈ π × [3(5 + 3) − √((15 + 3)(5 + 9))] = π × [24 − √252] ≈ π × 8.1255 ≈ 25.5269.
Sources: Ellipse — Wikipedia
Frequently asked questions
What are the semi-major and semi-minor axes?
The semi-major axis a is half of the ellipse’s longest diameter, and the semi-minor axis b is half of the shortest. They run perpendicular to each other through the centre, like a long and a short radius.
Why is the circumference only approximate?
The exact perimeter of an ellipse is an elliptic integral with no elementary formula. Ramanujan’s approximation used here is extremely close — error under about 0.04% for any ellipse — which is far tighter than most measurements need.
What happens when a equals b?
The ellipse becomes a circle of radius r = a = b. The area formula gives πr² and the circumference approximation returns exactly 2πr, matching the familiar circle results.
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This calculator is for educational purposes. Double-check important results before acting on them.