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Ellipse Eccentricity Calculator

Find the eccentricity of an ellipse.

How it is calculated

Enter the semi-major axis a (half of the longest diameter) and the semi-minor axis b (half of the shortest diameter), both in the same length unit. The tool computes e = √(1 − b²/a²), shown to four decimals, and the focal distance c = √(a² − b²), the distance from the centre to each focus, shown to two decimals. The focal distance is in whatever unit you used for the axes; the label only says 'units'. Both axes must be above zero and b cannot be larger than a: the tool does not swap them for you and shows no result if they are the wrong way round, or if either is zero or negative. When a equals b the shape is a circle and the eccentricity is 0.0000.

Worked example

Take an ellipse with a semi-major axis of 13 and a semi-minor axis of 5. Squares: a² = 169 and b² = 25. The ratio b²/a² is 25 ÷ 169 = 0.1479, so 1 − 0.1479 = 0.8521, and its square root is the eccentricity 0.9231. For the focal distance, 169 − 25 = 144 and √144 = 12.00 units. Each focus sits 12 from the centre, close to the end of the long axis at 13, which fits an elongated shape with e near 1.

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Common mistakes

  • Entering full axis lengths (diameters) instead of semi-axes. Halve them first. Eccentricity is unaffected because only the ratio b/a matters, but the focal distance would come out doubled.
  • Swapping the two values. The semi-major axis goes first and must be the larger one; if the smaller value is placed there, the tool returns no result.
  • Mixing units, for example a in metres and b in centimetres. Both axes must use the same unit or the eccentricity is meaningless.
  • Confusing the focal distance with the distance between the two foci. The tool gives the centre-to-focus value c; the distance between the foci is 2c.
  • Expecting an eccentricity of 1 or above. For a closed ellipse the value runs from 0 up to, but not including, 1.

About this calculator

Eccentricity measures how far an ellipse deviates from a perfect circle, computed as e = √(1 − b²/a²) from the semi-major axis a and semi-minor axis b; a value of 0 is a circle, and values approaching 1 describe an increasingly elongated ellipse. The same relationship gives the focal distance c = a·e, the distance from the ellipse's center to each of its two foci.

Astronomers use orbital eccentricity to characterize how elliptical a planet's, comet's or satellite's orbit is under Kepler's first law — Earth's orbit has e ≈ 0.017, nearly circular, while many comets have e close to 1. Mechanical engineers apply the same formula when designing elliptical gears and cams, and optical designers use it for elliptical mirrors and reflectors, which focus light or sound from one focus precisely to the other.

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