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

Find eccentricity from the two semi-axes.

Enter semi-axis lengths: half the full width or height. For an ellipse, a ≥ b > 0 and 0 ≤ e < 1.

Related Calculators

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

Select the value to solve for. Enter a and b to find e, b and e to find a, or a and e to find b. Semi-axis lengths must be positive; a zero minor axis is a degenerate line. A semi-axis is half the full axis length.

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

The descriptive ranges below are illustrative shape descriptions, not standardized classifications.

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.