Circle and Ellipse Area Calculator

Area of a circle and an ellipse

A circle is just the set of points that sit the same distance from a center, and its area works out to A = π·r². The ellipse is the more general shape: it has two semi-axes, a (the major) and b (the minor), so the area becomes A = π·a·b. Set a equal to b and you're back to a circle. How stretched the ellipse looks comes from its eccentricity, e = √(1 − b²/a²), which runs from 0 for a circle up toward 1 for something very elongated. Plug in r = 5 and you get A = 25π ≈ 78.540; with a = 4 and b = 3 the area is 12π ≈ 37.699.

Applications

Athletics tracks pair long straights with semicircular or elliptical ends, and you need their area to budget turf and surfacing. Planetary orbits trace ellipses, per Kepler's first law, while the area a planet sweeps out in a given time stays constant (that's the second law). The same geometry shows up in elliptical billiards and whispering-gallery acoustic chambers, where rays bounce from one focus to the other. Medical lithotripsy borrows the trick, focusing shock waves on a single point to break up kidney stones.

FAQ

Why is the area π·a·b and not π·(a+b)/2 squared? Picture a unit circle stretched by a along the x-axis and by b along the y-axis. That's an affine map, and area under it grows by the determinant a·b, so you land on π·a·b.

What is eccentricity used for? It puts a number on how stretched the ellipse is, which matters when you're working with orbital mechanics, optics or engineering tolerances.

Does this formula work in any unit? It does. Whatever linear unit you feed into r, a and b, the answer comes back squared (m → m², cm → cm²).