Calculate ellipse eccentricity, semi-major axis, or semi-minor axis from any two values, with length units in meters, feet, inches, or kilometers.

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 eccentricity of an ellipse measures how stretched the ellipse is compared with a circle. For an ellipse, eccentricity is always greater than or equal to 0 and less than 1.

e = √(1 - (b²) / (a²))
a = (b) / (√(1 - e²))
b = a√(1 - e²)
  • e = eccentricity of the ellipse, dimensionless
  • a = length of the semi-major axis
  • b = length of the semi-minor axis

If you enter a and b, the calculator finds e using the first formula. If you enter b and e, it solves for the semi-major axis a. If you enter a and e, it solves for the semi-minor axis b.

The calculator also converts length units before solving, so the semi-major and semi-minor axes can be entered in different supported units. Eccentricity has no unit.

Ellipse Eccentricity Values and Meaning

Eccentricity value Shape meaning
e = 0 A circle, where a = b
0 < e < 0.5 A slightly stretched ellipse
0.5 ≤ e < 0.9 A noticeably elongated ellipse
0.9 ≤ e < 1 A very long, narrow ellipse
e ≥ 1 Not an ellipse

Supported Length Unit Conversions

Unit Conversion to meters
Meter, m 1 m
Foot, ft 0.3048 m
Inch, in 0.0254 m
Kilometer, km 1000 m

Example Problems

Example 1: Find eccentricity from the axes

You have an ellipse with a semi-major axis of 10 m and a semi-minor axis of 8 m.

e = √(1 - (8²) / (10²))
e = √(1 - 0.64) = √(0.36) = 0.6

The eccentricity is 0.6000.

Example 2: Find the semi-minor axis

You have an ellipse with a semi-major axis of 12 ft and an eccentricity of 0.5.

b = 12√(1 - 0.5²)
b = 12√(0.75) ≈ 10.3923

The semi-minor axis is approximately 10.3923 ft.

FAQ

What is the eccentricity of a circle?

The eccentricity of a circle is 0. A circle is a special case of an ellipse where the semi-major axis and semi-minor axis are equal, so a = b.

Why must the semi-major axis be greater than or equal to the semi-minor axis?

In an ellipse, the semi-major axis is the longer radius from the center, and the semi-minor axis is the shorter radius. That means a ≥ b. If b is greater than a, the axes have been labeled in the wrong order.

Can ellipse eccentricity be 1?

No. For an ellipse, eccentricity must satisfy 0 ≤ e < 1. When eccentricity reaches 1, the shape is no longer an ellipse.