Eccentricity of an Ellipse: Explained with Examples

The eccentricity of an ellipse is a single number, written e, that measures how stretched the ellipse is. A nearly circular ellipse has e close to 0; a long, thin one has e close to 1. For x²/a² + y²/b² = 1 with a > b, it is defined as e = c/a, where c is the distance from the centre to each focus. The value of e always lies strictly between 0 and 1.

Eccentricity matters because it compares ellipse shapes at a glance, without drawing them. Astronomers use it for orbits: Earth's orbit has e ≈ 0.0167, so it looks almost perfectly circular, while some comets have e near 0.9. In exams, nearly every ellipse problem asks for e first, since the foci, vertices and latus rectum all depend on it.

Key Concepts

• Semi-major axis (a): the longest radius, from the centre to a vertex on the major axis.
• Semi-minor axis (b): the shortest radius, from the centre to a co-vertex; in the standard form, a is always the larger one.
• Focal distance (c): the distance from the centre to each focus, found from c² = a² − b².
• Eccentricity (e): defined as e = c/a; dimensionless, and always 0 < e < 1 for an ellipse.
• Circle as a special case: e = 0 merges the two foci at the centre (c = 0).

Formulas

• Definition: e = c/a, with c from c² = a² − b².
• Direct formula: e = √(1 − b²/a²), handy when c is unknown.
• Units: a, b, c are lengths (for example, metres); e is just a number.
• Assumptions: standard ellipse centred at the origin, a > b. If the larger denominator is under y², the major axis is vertical, but a is still the larger value.

Worked Example

Find the eccentricity of 25x² + 16y² = 400.

1. Standard form. Divide by 400: x²/16 + y²/25 = 1. The larger denominator is 25, so a² = 25, b² = 16; hence a = 5, b = 4.
2. Find c. c² = a² − b² = 25 − 16 = 9, so c = 3.
3. Find e. e = c/a = 3/5 = 0.6.

Answer: e = 0.6, a noticeably stretched ellipse roughly halfway between a circle and a very flat one.

Common Misconceptions

• "A very stretched ellipse has e > 1." Wrong — an ellipse always has 0 < e < 1; e above 1 means a hyperbola.
• "c² = a² + b²." That plus-sign formula is the hyperbola's; the ellipse subtracts: c² = a² − b².
• "Bigger e means rounder." The opposite: e near 0 is round, e near 1 is long and flat.

Real-World Applications

• Orbits: Kepler's orbits are ellipses; e shows how far each deviates from circular.
• Whispering galleries: in elliptical domes like St Paul's Cathedral, London, a whisper at one focus is heard clearly at the other.
• Elliptical gears: machines use them to vary rotation speed within each turn, with e setting the variation.
• Architecture: elliptical arches and running tracks use a chosen e to balance strength and space.

In short, e = c/a captures an ellipse's shape in one number between 0 and 1. Remember the minus sign in c² = a² − b², and the foci, vertices and tangents all follow easily.