Ellipse Circumference Calculator

Ellipse Circumference

Unlike a circle, an ellipse has no simple closed-form perimeter formula. The circumference involves an elliptic integral. Practical engineering uses accurate approximations such as Ramanujan's formulas.

Ramanujan II Approximation

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

where $h = \left(\frac{a - b}{a + b}\right)^2$, $a$ is the semi-major axis, and $b$ is the semi-minor axis. This formula typically has error below 0.0001%.

Ramanujan I Approximation

$$C \approx \pi\left[3(a + b) - \sqrt{(3a + b)(a + 3b)}\right]$$

When $a = b$, both formulas reduce to the circle circumference $C = 2\pi a$.

Related tools: Ellipse Calculator and Area Calculator.

Frequently Asked Questions

Why is there no exact formula for ellipse circumference?

The perimeter requires a complete elliptic integral of the second kind, which cannot be expressed with elementary functions like polynomials or standard trig functions.

Which formula is most accurate?

Ramanujan II is the most accurate simple approximation for nearly all practical ellipse shapes. This tool also compares Ramanujan I, a simple formula, and a series expansion.

What if my semi-major axis is smaller than semi-minor?

Enter the longer half-axis as $a$ and the shorter as $b$. The calculator uses the values you provide and computes eccentricity accordingly.

How is eccentricity calculated?

Eccentricity $e = \sqrt{1 - b^2/a^2}$ when $a \geq b$. It measures how elongated the ellipse is, from 0 (circle) toward 1 (very elongated).

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Ellipse Circumference Calculator