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Conic Section Eccentricity Calculator

Choose the conic type and enter the two semi-axes to compute eccentricity.

How it is calculated

You give two positive lengths, a and b, and a conic type. For a hyperbola the tool uses c = √(a² + b²) and e = c / a, so a is the semi-transverse axis. For an ellipse it treats the larger of the two numbers as the semi-major axis and the smaller as the semi-minor axis, so the order you type them in does not matter: c = √(major² − minor²) and e = c / major. Both results are shown to six decimals. Zero, negative or empty values return no result. Use the same length unit for both axes; eccentricity has no unit, while the focal distance c comes out in that unit.

Worked example

Take a hyperbola with a = 3 and b = 4. Step 1: a² + b² = 9 + 16 = 25. Step 2: c = √25 = 5, the distance from the centre to each focus. Step 3: e = c / a = 5 / 3. The tool reports an eccentricity of 1.666667 and a focal distance of 5.000000. Since e is above 1, this is consistent with a hyperbola; an ellipse always gives a value below 1.

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Common mistakes

  • Typing full axis lengths (major and minor diameters) instead of semi-axes, which halves the real values — the ratio stays the same for an ellipse, but the focal distance c is doubled.
  • Leaving the type on ellipse when the curve is a hyperbola: for a = 3, b = 4 the ellipse setting gives 0.661438 instead of 1.666667.
  • Assuming that for a hyperbola the larger number is the major axis; here a is always the transverse semi-axis, so swapping a and b changes the answer.
  • Mixing units, such as a in metres and b in centimetres, which gives a meaningless ratio.
  • Expecting a parabola option: only ellipse and hyperbola are handled, since a parabola's eccentricity is fixed at 1.

About this calculator

Eccentricity describes how much a conic section deviates from a perfect circle. For an ellipse with semi-major axis a and semi-minor axis b, e = √(1 − b²/a²), giving a value between 0 (a circle) and 1 (nearly a line); for a hyperbola, e = √(1 + b²/a²), always greater than 1. In both cases the focal distance from the center is c = a·e, which is what actually defines the position of the foci.

Astronomers and orbital mechanics engineers use eccentricity to characterize how elongated a planet's, comet's or satellite's orbit is — Earth's orbit is nearly circular at e ≈ 0.017 while many comets have e close to 1. Optical engineers rely on the same math when designing elliptical or hyperbolic reflectors and lenses, where a conic's eccentricity determines how it focuses light or radio waves.

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