Eccentricity of an Ellipse Calculator

A simple and powerful tool to understand the formula used to calculate the eccentricity of an ellipse, a key parameter in geometry and astronomy.

Visual representation of the ellipse with its axes and foci.

What is the Eccentricity of an Ellipse?

The eccentricity of an ellipse, denoted by the letter ‘e’, is a fundamental parameter that measures how much the ellipse deviates from being a perfect circle. It’s a non-negative real number that provides a numerical value for its “flatness.” The formula used to calculate the eccentricity of an ellipse defines this shape characteristic precisely. An eccentricity of 0 corresponds to a circle. As the eccentricity increases towards 1, the ellipse becomes more elongated or “squashed.”

This concept is crucial not only in geometry but also in physics and astronomy. For instance, the orbits of planets, asteroids, and comets around the Sun are ellipses. [8] Understanding the eccentricity of these orbits helps scientists predict their paths and behaviors. Anyone from a math student to an aerospace engineer might use this calculation.

Formula Used to Calculate the Eccentricity of an Ellipse

There are a couple of primary formulas to determine the eccentricity. The most common one, and the one this calculator uses, relates the semi-major and semi-minor axes of the ellipse.

The primary formula is:

e = √(1 – (b² / a²))

Another related formula uses the linear eccentricity (c), which is the distance from the center to a focus. [4]

e = c / a

Where c itself is calculated as: c = √(a² – b²)

Variables in the Eccentricity Formula

Variable	Meaning	Unit (Inferred)	Typical Range
e	Eccentricity	Unitless	0 ≤ e < 1
a	Semi-major Axis	Length (km, m, px, etc.)	a > 0
b	Semi-minor Axis	Length (km, m, px, etc.)	0 < b ≤ a
c	Linear Eccentricity (Center to Focus)	Length (km, m, px, etc.)	0 ≤ c < a

Practical Examples

Example 1: A Nearly Circular Orbit

Imagine a planet with a very stable, nearly circular orbit. Its orbital parameters might be:

• Inputs:
  • Semi-major Axis (a): 150,000,000 km
  • Semi-minor Axis (b): 149,000,000 km
• Results:
  • Using the formula used to calculate the eccentricity of an ellipse, e = √(1 – (149M² / 150M²)) ≈ 0.115
  • This low eccentricity value confirms the orbit is very close to a circle.

Example 2: A Highly Elliptical Comet Path

Now consider a comet with a very elongated path, like Halley’s Comet.

• Inputs:
  • Semi-major Axis (a): 17.8 AU (Astronomical Units)
  • Semi-minor Axis (b): 4.5 AU
• Results:
  • e = √(1 – (4.5² / 17.8²)) ≈ 0.967
  • This value is very close to 1, indicating a very stretched-out, flat ellipse, which is typical for many comets. For more on orbital mechanics, you might want to explore our orbital period calculator.

How to Use This Eccentricity Calculator

1. Enter the Semi-major Axis (a): Input the length of the longest radius of your ellipse into the first field. This value must be a positive number.
2. Enter the Semi-minor Axis (b): Input the length of the shortest radius. This value must be positive and less than or equal to the semi-major axis. The calculator will show an error if b > a.
3. Select Units: Choose the appropriate unit of measurement for your axes from the dropdown menu. This choice affects the labels for intermediate results but not the unitless eccentricity value itself.
4. Interpret the Results: The calculator automatically updates, showing the final eccentricity (e). A value near 0 means your ellipse is almost a circle. A value near 1 means it’s very elongated. You can also see intermediate values like the linear eccentricity and the total area of the ellipse.
5. Visualize: The dynamic chart below the results provides a visual representation of your ellipse, updating in real-time as you change the inputs.

Key Factors That Affect Ellipse Eccentricity

• Ratio of a/b: The single most important factor is the ratio between the semi-major (a) and semi-minor (b) axes. The closer ‘b’ is to ‘a’, the closer the ratio b/a is to 1, and the closer the eccentricity gets to 0.
• Focal Distance: The distance between the two foci (2c) directly influences eccentricity. If the foci are far apart (large ‘c’), the eccentricity is high. If they are close together (small ‘c’), the eccentricity is low.
• Orbital Energy (Physics): In orbital mechanics, a body’s eccentricity is determined by its total energy. Lower energy orbits for a given system are more circular.
• Initial Velocity and Angle (Physics): When an object enters an orbit, its initial velocity and angle of approach relative to the central body determine the shape (and thus eccentricity) of its resulting elliptical path.
• Gravitational Perturbations: The orbits of planets are not perfect ellipses because of gravitational tugs from other planets. These perturbations can cause the eccentricity of an orbit to change slowly over thousands of years.
• Measurement Units: The choice of units (km, miles, AU) for ‘a’ and ‘b’ does not affect the final eccentricity value, as it is a dimensionless ratio. However, consistency is key; ‘a’ and ‘b’ must be in the same units.

Frequently Asked Questions (FAQ)

What is the eccentricity of a perfect circle?

A perfect circle has an eccentricity of 0. This occurs when the semi-major axis (a) and semi-minor axis (b) are equal. In this case, the foci merge at the center (c=0), and the formula used to calculate the eccentricity of an ellipse yields e = 0.

Can the eccentricity of an ellipse be 1 or greater?

No. By definition, the eccentricity of an ellipse is always greater than or equal to 0 and strictly less than 1 (0 ≤ e < 1). An eccentricity of exactly 1 defines a parabola, and a value greater than 1 defines a hyperbola. See our conic sections overview for more.

What are the units of eccentricity?

Eccentricity is a dimensionless (or unitless) quantity. It is a pure ratio of two lengths (e.g., e = c/a), so the units cancel out.

Why is my semi-minor axis input showing an error?

The semi-major axis ‘a’ must, by definition, be greater than or equal to the semi-minor axis ‘b’. If you enter a value for ‘b’ that is larger than ‘a’, the calculator will flag it as an invalid input for an ellipse.

What is linear eccentricity?

Linear eccentricity, denoted by ‘c’, is the distance from the center of the ellipse to either of its two foci. It’s calculated with the formula c = √(a² – b²). [9]

How does changing the units affect the calculation?

Changing the units in the selector only changes the labels for the intermediate results (like Area and Linear Eccentricity). The main result, the eccentricity ‘e’, is a ratio and remains the same regardless of the unit system used, as long as both axes use the same unit.

Where are ellipses found in the real world?

Ellipses are everywhere! The orbits of planets, moons, and satellites are ellipses. The shape of many gears in machines can be elliptical. When you tilt a glass of water, the surface of the water forms an ellipse. Even a circle viewed from an angle appears as an ellipse. [8]

What’s the difference between the major axis and the semi-major axis?

The major axis is the longest diameter of the ellipse, passing through the center and both foci. The semi-major axis (‘a’) is half of that length—from the center to one of the two farthest points on the ellipse. The same relationship holds for the minor and semi-minor axes.