Pick the unit your lengths are in. Areas are then labelled in square units and volumes in cubic units; nothing is converted.
- P ≈ π · (3(a + b) − √((3a + b)(a + 3b)))
a and b are the semi-axes, half the longest and half the shortest width. The formula is Ramanujan’s first approximation from 1914, cited under the formula; it is not exact, and no formula with finitely many roots and products is, because the exact perimeter is an elliptic integral.
Holds when the ellipse is not extremely flat. For b / a down to about 0.2 the error is below 0.1%; as b approaches 0 the formula underestimates, by 0.4% at the limit.
A worked example
Semi-axes of 5 and 3 cm: 3(5 + 3) = 24, (15 + 3)(5 + 9) = 18 · 14 = 252, √252 = 15.87, and P ≈ π · (24 − 15.87) = π · 8.13 ≈ 25.5 cm. The exact perimeter, from the integral, is 25.527 cm; the approximation is off by less than a hundredth of a per cent.
The circle as a check
With a = b = r the bracket is 6r − √(4r · 4r) = 6r − 4r = 2r, and P = 2πr: the circumference of a circle comes out exactly. Any proposed ellipse formula has to pass that test, and the common wrong one, π(a + b), passes it too and then drifts to 10% low for a flat ellipse, which is why Ramanujan’s is the one worth remembering.
The area is πab, exact and simple, because stretching a circle scales areas by a constant. The perimeter does not scale that way: a stretched circle’s edge is stretched more in some places than others, and summing it gives an integral with no closed form.
Common mistakes
- Using π(a + b). Right for a circle, 10% low for a 10 : 1 ellipse.
- Using the full axes. a and b are halves, as in the area formula.
- Reading the result as exact. It is a very good approximation; the page says so.
Frequently asked questions
Is there a better approximation?
Ramanujan’s second, with h = ((a − b) / (a + b))², is better still; this one is accurate enough for
anything measured with a tape.
Where does the exact value come from?
The complete elliptic integral of the second kind, 4a · E(e), evaluated numerically. A calculator with
the function gives it; a formula does not.
What is a and what is b?
a is the semi-major axis, the larger. The formula is symmetric, so swapping them changes nothing.