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.

5 Fun Facts about Ellipses

Stretch a circle, get an ellipse

An ellipse is just a circle viewed through an affine stretch. Any circle formula survives the trip if you scale one axis—no rotation needed.

Affine twin

Laser whispers

Sound or light launched from one focus reflects to the other. That’s why whispering galleries and elliptical billiards feel spooky-precise.

Focus magic

Kepler’s almost-circle

Earth’s orbit is an ellipse, but its eccentricity is only ~0.0167. If you drew it to scale, you’d squint to see it isn’t a circle.

Space subtlety

Perimeter resists closed form

Ramanujan’s 1914 approximations nail ellipse perimeter to many decimals without elliptic integrals—handy when you need speed plus surprising accuracy.

Ramanujan win

Sum-of-distances constant

Every point on an ellipse keeps PF₁ + PF₂ constant (equal to 2a). It’s the string-and-pins trick behind the classic boardroom-table demo.

Geometry anchor

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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