Major Axis — Definition, Formula & Examples

The major axis is the longest line segment that passes through the center of an ellipse, connecting the two endpoints (vertices) that are farthest apart. It always passes through both foci of the ellipse.

For an ellipse with semi-major axis length $a$ and semi-minor axis length $b$ where $a > b$ , the major axis is the chord of length $2a$ that lies along the axis containing both foci and intersects the ellipse at its two vertices.

Key Formula

\text{Length of major axis} = 2a

Where:

$a$ = The semi-major axis — the distance from the center of the ellipse to either vertex along the major axis

Worked Example

Problem: Find the length of the major axis for the ellipse given by the equation

\frac{x^2}{25} + \frac{y^2}{9} = 1

Identify a²: The standard form of an ellipse is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where

a > b

. Here, the larger denominator is 25, so

a^2 = 25

a^2 = 25

Find a: Take the square root to get the semi-major axis length.

a = \sqrt{25} = 5

Calculate the major axis length: The full major axis is twice the semi-major axis.

2a = 2(5) = 10

Answer: The major axis has a length of 10 units and lies along the

x

-axis, from

(-5, 0)

(5, 0)

Why It Matters

The major axis determines the orientation and overall shape of an ellipse — whether it stretches horizontally or vertically. Orbital mechanics uses it directly: Kepler's first law states planets travel in ellipses, and the major axis defines the longest dimension of each orbit.

Common Mistakes

Mistake: Confusing the major axis with the semi-major axis and using

a

instead of

2a

for its length.

Correction: The semi-major axis is

a

(center to vertex), while the full major axis spans both vertices and has length

2a