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Eccentricity an Ellipse
A measure of how 'out of round' an ellipse is. It is given by the formula where c is the distance from the center to a focus and
a is the distance from that focus to a vertex
Try this Drag the orange dots to resize the ellipse. As the shape and size of the ellipse changes, the eccentricity is recalculated.

If you think of an ellipse as a 'squashed' circle, the eccentricity of the ellipse gives a measure of just how 'squashed' it is. It is found by a formula that uses two measures of the ellipse. where
c is the distance from the center to a focus.
a is the distance from that focus to a vertex

The formula produces a number in the range 0..1    If the eccentricity is zero, it is not squashed at all and so remains a circle. If it is 1, it is completely squashed and looks like a line. In the applet above, drag the orange dots to create both these eccentricities and some in between.

The word "eccentric"

The word means "off center". It is probably used because the more eccentric an ellipse is, the more its foci are 'off the center' of the ellipse. Kepler discovered in the 1500's that planets are often in "eccentric orbits" instead of exact circles. These orbits turned out to be ellipses with the sun at one of the focus points. In this context, the eccentricity of the ellipse indicates how far from circular these orbits are.

Textbook definition

Many textbooks define eccentricity as how 'round' the ellipse is. Since the value increases as the ellipse is more "squashed", this seems backwards. For that reason it is described here as how out of round,or squashed, it is.

Try this

  1. In the figure above, click on 'reset' and 'hide details'
  2. Drag one of the orange dots on the edge of the ellipse to make a random size ellipse.
  3. Calculate the eccentricity of the ellipse
  4. Click 'Show details' to check your answer.

Other ellipse topics