Enter the length of the semi-major axis and semi-minor axis into the calculator to determine the eccentricity (e) of an ellipse. This calculator can also evaluate any of the variables given the others are known.

Ellipse Eccentricity Calculator

Enter exactly 2 values to calculate the missing variable (for an ellipse, a ≥ b and 0 ≤ e < 1)

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Ellipse Eccentricity Formula

The following formula is used to calculate the eccentricity (e) of an ellipse from its semi-major axis (a) and semi-minor axis (b).

e = \sqrt(1 - (b^2 / a^2))

Variables:

  • e is the eccentricity of the ellipse (dimensionless)
  • a is the length of the semi-major axis
  • b is the length of the semi-minor axis

To calculate the eccentricity, square the length of the semi-minor axis and divide it by the square of the length of the semi-major axis. Subtract this result from 1 and then take the square root of the result. For an ellipse, a must be greater than or equal to b, and the resulting eccentricity satisfies 0 ≤ e < 1 (with e = 0 being a circle).

What is Ellipse Eccentricity?

Ellipse eccentricity (e) is a dimensionless measure of how “stretched” an ellipse is compared to a circle. It can be defined as e = c/a, where c is the distance from the center of the ellipse to either focus and a is the semi-major axis. Using the relationship between the semi-axes, it can also be computed as e = √(1 − b²/a²). Values range from 0 (a perfect circle) up to but not including 1 (an increasingly elongated ellipse).

How to Calculate Ellipse Eccentricity

The following steps outline how to calculate ellipse eccentricity (e).

  1. First, determine the length of the semi-major axis (a).
  2. Next, determine the length of the semi-minor axis (b), ensuring a ≥ b.
  3. Next, gather the formula from above: e = √(1 - (b² / a²)).
  4. Finally, calculate the eccentricity (e).
  5. After inserting the values of a and b into the formula and calculating the result, check your answer with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Length of the semi-major axis (a) = 5

Length of the semi-minor axis (b) = 3

Then e = √(1 - (3² / 5²)) = √(1 - 9/25) = √(16/25) = 0.8.