An Ellipse Has a Simple Area and No Simple Perimeter

The approximation degrades too, just far later

At semi-axes of 5 and 3 the calculator's answer and the exact integral agree to six figures. Stretch the shape to 10 and 1 and the agreement loosens to about eight parts in ten thousand — still far better than anything a tape measure would resolve, and no longer the exactness the displayed digits imply. That is the honest reading of any figure produced by an approximation: it is right to a stated precision that depends on the inputs, not right absolutely. Where an ellipse is close to circular the approximation is essentially perfect; where it is a long thin sliver, quote fewer digits than the screen offers.

Where the problem actually shows up

Anywhere a length has to be cut to fit an oval. The trim around an elliptical mirror, the edging of an oval bed, the belt around two unequal pulleys and the seam of an oval tabletop all need the perimeter and none of them are helped by the area. The historical case is planetary orbits: the distance a planet travels in one circuit is exactly this integral, and the family of functions invented to handle it took its name from the problem. A practical rule when the shape is drawn rather than specified: measure the outline with a flexible tape rather than computing it, because a real oval is rarely a mathematical ellipse in the first place.

Frequently asked questions

What does it mean that no formula exists?

It is a statement about a specific vocabulary, not an admission of ignorance. The perimeter is perfectly well defined and can be computed to as many digits as anyone wants; what it cannot be is written down using only the familiar operations — roots, powers, logarithms, trigonometric functions and their combinations. Mathematicians proved that no such expression exists, then defined new functions specifically to name the quantity, and those functions are now as computable as a sine. The circle is the exception, not the rule: most curves have lengths that behave like this one, and the circle's tidy 2πr is a piece of good luck rather than the normal state of affairs.

Does setting both axes equal really give a circle?

Yes, and it is the fastest available sanity check on any ellipse formula. Semi-axes of 5 and 5 give a perimeter of 31.4159, which is 2π times 5 to every digit shown, and the area formula collapses to πr² at the same moment. Any approximation worth using reproduces the circle exactly rather than approximately, because the awkward part of the integral vanishes when the two axes match. An expression that got the circle slightly wrong would be telling you it is doing something other than what it claims — which makes this the first test to run on a formula found in a forum post.