The area is exactly πab. A 5-by-3 ellipse has area 15π ≈ 47.12. The perimeter has no exact elementary formula — it is an elliptic integral — so every figure quoted for it, here included, is an approximation.
Area is the easy half
πab. Exact, simple, and a clean generalisation of a circle’s πr².
It works because an ellipse is a circle that has been stretched by a/r in one direction and b/r in the other, and stretching scales area by the product of the two factors.
That same argument is why the area formula is easy and the perimeter formula is impossible. Stretching multiplies areas by a constant; it does not multiply lengths by a constant, because different parts of the curve run in different directions and are stretched by different amounts.
The perimeter problem
There is no elementary formula for the perimeter of an ellipse. Not one that is hard to find — one that does not exist.
The exact value is a complete elliptic integral of the second kind, and it cannot be written with the usual functions: no combination of roots, logarithms, exponentials or trigonometry produces it. The whole theory of elliptic functions grew out of this problem.
So every quoted formula is an approximation. The common ones:
π(a + b) is the crudest, and it is only right for a circle. π√(2(a² + b²)) is better and still poor. Ramanujan’s second approximation, which this page uses, is remarkably good — accurate to about a part in ten billion for a moderate ellipse.
But it degrades. As the ellipse elongates the error grows, reaching about four parts in ten thousand for a 1000:1 shape. This page computes a converged series alongside it and reports the gap, because a reader working with an elongated ellipse deserves to know rather than to assume.
Eccentricity
One number for how far from circular an ellipse is. Zero is a perfect circle; approaching one is a flattened sliver.
e = √(1 − b²/a²), and it is also the ratio of the focal distance to the semi-major axis.
It is worth calibrating against real numbers, because eccentricity is smaller than it looks. Earth’s orbit has e = 0.0167, which is why every scale drawing of it looks circular — the difference between its axes is about a hundredth of a percent. Mars is 0.093, Pluto 0.249, and Halley’s comet 0.967.
Eccentricity classifies all the conics: 0 is a circle, between 0 and 1 an ellipse, exactly 1 a parabola, and above 1 a hyperbola. One number, four curves.
The two foci
Two special points inside, at ±ae from the centre along the long axis, and the ellipse is defined by them.
Every point on the curve has the same total distance to the two foci. That is what an ellipse is — not a stretched circle, though it is also that.
The definition is directly practical. Two pins and a loop of string longer than the gap between them, pulled taut by a pencil, draws an ellipse. Gardeners use exactly this to lay out oval beds.
The reflective property follows: anything leaving one focus arrives at the other. It is why a whispering gallery works, and why an elliptical reflector can focus shock waves onto a kidney stone from outside the body.
Kepler’s first law says the planets travel on ellipses with the Sun at one focus. The other focus is empty — there is nothing there at all.
Where it gets used
Orbits. Every planet, moon, comet and satellite follows one. The eccentricity is what distinguishes a near-circular orbit from a long looping one.
Optics and acoustics. Elliptical reflectors focus energy from one point to another, in whispering galleries and in medical lithotripsy.
Engineering. Elliptical pipe sections, oval tanks and the cross-sections of aircraft fuselages. The area formula is exact and gets used constantly; the perimeter one is approximate and gets used carefully.
Statistics. Confidence regions for two correlated variables are ellipses, and their axes are the eigenvectors of the covariance matrix.
Sources and methodology
The area, the elliptic integral and Ramanujan’s approximation are standard; these are the references.
Method. The area is πab and is exact. The perimeter is computed two ways: Ramanujan’s second approximation, which is what the page reports as the answer, and a Gauss-Kummer series taken to convergence, which serves as the reference. The relative difference between them is displayed rather than hidden, because that difference grows to about four parts in ten thousand for a very elongated ellipse — visible, and worth knowing. The suite checks the series itself against numerical integration of the elliptic integral, since a wrong reference would report a false error figure and that is worse than reporting none. That engine is verified on every change against 114 hand-written assertions, including that the series reference matches direct numerical integration of the elliptic integral, and that Ramanujan’s approximation is exact for a circle and demonstrably degrades as the ellipse elongates. The count and the per-case breakdown are published on the formula verification page.
Read the guide
A circle is the one ellipse whose perimeter does have an exact formula — the Circle Calculator handles that case.