Ellipse Calculator

Frequently Asked Questions

Should I enter semi-axes or full diameters?

Enter the semi-axes, the distances from the center to the edge. The full width and height are twice these values.

How is the circumference calculated?

An ellipse has no simple exact perimeter, so the calculator uses the Ramanujan approximation C ≈ π[3(a+b) − √((3a+b)(a+3b))], accurate to a tiny fraction of a percent.

What does eccentricity mean?

Eccentricity e = √(1 − b²/a²) ranges from 0 for a circle toward 1 for a long, thin ellipse, measuring how stretched the shape is.

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Important Disclaimer: Estimates for informational purposes only.

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How This Calculator Works

An ellipse is the stretched circle traced by planetary orbits and the cross sections of many engineered shapes. With semi-major axis a and semi-minor axis b, the area is exactly A = πab. The perimeter has no simple closed form, so this calculator uses the Ramanujan approximation C ≈ π[3(a+b) − √((3a+b)(a+3b))], accurate to a tiny fraction of a percent for ordinary ellipses. The eccentricity e = √(1 − b²÷a²) measures how far the shape departs from a perfect circle.

How To Use It

Enter the two semi-axes - the distances from the center to the edge along the long and short directions. The order does not matter; the calculator sorts them so the larger becomes the semi-major axis a and the smaller becomes the semi-minor axis b. It returns the area A = πab, the Ramanujan circumference, and the eccentricity e.

Reading The Eccentricity

Eccentricity ranges from 0 to just under 1. A value of 0 means a perfect circle where both axes are equal. As the ellipse stretches, e climbs toward 1, where the shape becomes long and thin. Earth's orbit has e ≈ 0.0167 - nearly circular - while many comets have e > 0.9.

Tips

Enter semi-axes, not full diameters: a full width of 10 means a semi-axis of 5.

Equal axes (a = b = r) give a circle with A = πr², C = 2πr, and e = 0.

The Ramanujan perimeter is an approximation, accurate but not exact; relative error is below 10⁻⁵ for e ≤ 0.95.

Ellipse Calculator

Frequently Asked Questions

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