Perimeter of an Ellipse — Definition, Formula & Examples

The perimeter of an ellipse is the total distance around its curved boundary. Unlike a circle, there is no simple exact formula, so approximations are used.

The perimeter (circumference) of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is given by the elliptic integral $P = 4a\int_0^{\pi/2}\sqrt{1 - e^2\sin^2\theta}\,d\theta$ , where $e$ is the eccentricity. Since this integral has no closed-form solution in terms of elementary functions, various approximation formulas are employed in practice.

Key Formula

P \approx \pi(a + b)\left(1 + \frac{3h}{10 + \sqrt{4 - 3h}}\right)

Where:

$P$ = Approximate perimeter of the ellipse
$a$ = Semi-major axis (half the longer diameter)
$b$ = Semi-minor axis (half the shorter diameter)
$h$ = Defined as h = (a − b)² / (a + b)²

How It Works

The most popular approximation is Ramanujan's formula, which is accurate to within about 0.04% for most ellipses. You only need the semi-major axis

a

(half the longest diameter) and the semi-minor axis

b

(half the shortest diameter). Compute

h = \frac{(a - b)^2}{(a + b)^2}

, then plug into the formula. When

a = b

, the ellipse becomes a circle and the formula simplifies to

2\pi r

, as expected.

Worked Example

Problem: Find the approximate perimeter of an ellipse with semi-major axis a = 5 and semi-minor axis b = 3.

Compute h: Calculate h using the two semi-axes.

h = \frac{(5 - 3)^2}{(5 + 3)^2} = \frac{4}{64} = 0.0625

Evaluate the correction factor: Substitute h into the denominator and fraction inside the formula.

10 + \sqrt{4 - 3(0.0625)} = 10 + \sqrt{3.8125} \approx 10 + 1.9526 = 11.9526

Apply Ramanujan's formula: Multiply through to get the perimeter.

P \approx \pi(5 + 3)\left(1 + \frac{3(0.0625)}{11.9526}\right) = 8\pi\left(1 + 0.01569\right) \approx 8\pi(1.01569) \approx 25.53

Answer: The perimeter is approximately 25.53 units.

Why It Matters

Elliptical perimeters arise in orbital mechanics (planetary orbits are ellipses), engineering design of oval tracks, and architecture. Knowing a reliable approximation saves you from evaluating difficult integrals by hand.

Common Mistakes

Mistake: Using the circle formula 2πr with the average of a and b, i.e., P = 2π·(a+b)/2 = π(a+b).

Correction: This simpler formula underestimates the perimeter whenever a ≠ b. Use Ramanujan's approximation, which adds the correction factor involving h to account for the ellipse's elongation.