Ellipse Area Calculator
Calculate the area of an ellipse from its semi-major and semi-minor axes. Formula: A = π × a × b.
Formula
A = π × a × b
Area of an ellipse
An ellipse is the set of points where the distances to two foci always add up to the same total. Call the semi-major axis a and the semi-minor axis b, and the area is A = π·a·b. So with a = 5 m and b = 3 m you get A = π · 5 · 3 ≈ 47.12 m². Set a = b = r and the ellipse turns into a plain circle, with the formula collapsing to A = π·r². Eccentricity, e = √(1 - b²/a²), tells you how round it is: e = 0 is a circle, while e → 1 gives a long, stretched, almost parabola-shaped ellipse.
The perimeter is a different story. There's no closed elementary formula for it, so people fall back on approximations, and Ramanujan's is the favourite: P ≈ π[3(a+b) - √((3a+b)(a+3b))]. For most ellipses you'll ever deal with it stays within 0.01%. If you need the exact figure, that comes from the complete elliptic integral of the second kind.
Applications: from planetary orbits to medicine
Kepler worked out that planets move in elliptical orbits with the Sun at one focus, and the same curve shows up for comets, satellites, and electron orbitals in semiclassical models. Running tracks have semicircular ends linked by straightaways, yet stadium layouts and field events draw on genuine ellipses. Then there's the reflective property: a ray that leaves one focus bounces off the wall and lands on the other focus. That single fact underpins lithotripsy, in which shock waves aimed by an elliptical reflector break up kidney stones without any surgery, and it explains "whispering galleries", where a voice at one focus carries clearly across to someone standing at the other.
Calculate the area of an ellipse
An ellipse is roughly a "flattened" circle. Its area generalises the circle's, swapping pi times the radius squared for pi times the two semi-axes. The tool applies A = π × a × b from the major and minor semi-axes you enter.
It works for maths exercises, design projects, areas of oval shapes and engineering applications. Type the two semi-axes. Pi goes into the math on its own, so you don't need to recall the formula or the value of pi.
The calculation happens in your own browser, sending nothing out. A practical reference for an ellipse's area, a shape that shows up less than the circle but causes more confusion.
Frequently asked questions
What are semi-axes versus full axes?
How does the area compare with the bounding rectangle?
How to find the foci?
Is Earth's orbit a circle or an ellipse?
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.