Ellipse Eccentricity Calculator

Formula

e = √(1 − b²/a²)

Worked example

With Semi-major axis: 8 units, Semi-minor axis: 5 units, eccentricity is 0.780625.

Calculation method

The rule is e = √(1 − b²/a²).

For this ellipse eccentricity example, changing semi-major axis from 8 units to 9 units while keeping the other inputs fixed changes eccentricity from 0.780625 to 0.831479.

Reading eccentricity

Area measures the enclosed surface. Eccentricity is dimensionless: zero is a circle, while values closer to one describe a more elongated ellipse. In the worked values on this page, semi-major axis 8 units, semi-minor axis 5 units produce eccentricity 0.780625.

Practical use

Use these measures for elliptical layouts, optics, and geometry problems. The specific task here is to find eccentricity from semi-major axis, semi-minor axis.

Limits and common mistakes

A semiaxis is half the full width in that direction. For eccentricity, the semimajor axis must be at least as long as the semiminor axis. Verify semi-major axis in units before interpreting eccentricity; a value measured on another basis would answer a different question.

Why the relationship works

An ellipse is described here by its semimajor and semiminor axes, each measured from the center to the curve. Its area is π times their product. Eccentricity describes shape rather than size: zero is a circle, and a value nearer one is more elongated. The major semiaxis must be at least as long as the minor one. The ellipse eccentricity rule uses semi-major axis (units), semi-minor axis (units) in e = √(1 − b²/a²); read each symbol as the field named on this page.

Interpreting another set of values

For ellipse eccentricity, the fields labeled semi-major axis (units), semi-minor axis (units) are the known quantities. The calculator solves for eccentricity using e = √(1 − b²/a²). An ellipse area describes size, while eccentricity describes shape. Both depend on semiaxes, which are measured from the center rather than across the entire figure. When comparing two calculations, keep their units and input definitions identical.

For a second ellipse eccentricity check, keep the other default inputs fixed and change semi-major axis from 8 units to 10 units. Eccentricity changes from 0.780625 to 0.866025. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

The semimajor axis must be at least as long as the semiminor axis. The area is in square units, while eccentricity has no unit and lies from zero toward one for an ordinary ellipse. Semiaxes are half of the full widths; entering a full width instead doubles an axis. The reported eccentricity is 0.780625 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.