Ellipse Area & Perimeter Calculator

Enter semi-axes, full axes, area, or eccentricity. Private by design — runs locally in your browser.

Diagram & Inputs

Tip: Give any sensible combo. The solver prefers to deduce a and b first, then computes area and three perimeter values. If inputs conflict, it warns but still shows a best-fit result.

Results

Notes

Ellipse perimeter has no simple closed form. We show three values:

  • Ramanujan I: quick, good.
  • Ramanujan II: extremely accurate for all shapes.
  • Elliptic E (series): high-precision using the complete elliptic integral \(E(k)\) with \(k=e=\sqrt{1-b^2/a^2}\).

How the Ellipse Calculator Works

An ellipse is defined by its semi-major axis \(a\) and semi-minor axis \(b\). Area is simple: \(A=\pi ab\). Perimeter (circumference) has no elementary formula, so we compute trusted approximations (Ramanujan I & II) and a high-precision series for the complete elliptic integral \(E(k)\) with \(k=e\).

  • Axes: Major \(A=2a\), Minor \(B=2b\).
  • Eccentricity: \(e=\sqrt{1-b^2/a^2}\) (assumes \(a\ge b\)).
  • Ramanujan I: \(P\approx \pi\,[3(a+b)-\sqrt{(3a+b)(a+3b)}]\).
  • Ramanujan II: \(h=((a-b)^2/(a+b)^2)\), \(P\approx \pi(a+b)\left(1+\frac{3h}{10+\sqrt{4-3h}}\right)\).
  • Elliptic integral (series): \(P=4a\,E(e)\), with \(E(e)=\frac{\pi}{2}\sum_{n=0}^{\infty} c_n e^{2n}\) (we sum until tiny).

All computation is 100% client-side for privacy.

Ellipse Calculator: FAQs

Which inputs are valid?

Any of: \(a,b\); \(A=2a,B=2b\); area; eccentricity plus one semi-axis. Multiple inputs are OK; we check consistency.

Why multiple perimeters?

Ellipse perimeter lacks a simple exact formula. Ramanujan II is typically accurate to many decimals; the elliptic-integral series provides high precision.

What about units?

Axes and perimeter share your length unit; area uses squared units (e.g., cm²).

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