Semi-Major Axis — Definition, Formula & Examples

The semi-major axis is half the length of the longest diameter of an ellipse. It measures the distance from the center of the ellipse to the farthest point on its edge.

For an ellipse with major axis of length $2a$ , the semi-major axis is the value $a$ , equal to half the major axis. In the standard form equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$ , the semi-major axis $a$ extends along the $x$ -axis from the center to a vertex.

Key Formula

a = \frac{\text{major axis length}}{2}

Where:

$a$ = Semi-major axis (half the longest diameter)

How It Works

In the standard equation of an ellipse, the semi-major axis

a

always corresponds to the larger denominator under the squared terms. If the larger denominator sits under

x^2

, the major axis is horizontal; if it sits under

y^2

, the major axis is vertical. The semi-major axis also connects to the foci through the relationship

c^2 = a^2 - b^2

, where

c

is the distance from the center to each focus.

Worked Example

Problem: Find the semi-major axis and the distance to each focus for the ellipse

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

Identify a² and b²: The larger denominator is 25 and the smaller is 9, so

a^2 = 25

and

b^2 = 9

a^2 = 25, \quad b^2 = 9

Find the semi-major axis: Take the square root of

a^2

a = \sqrt{25} = 5

Find the focal distance: Use

c^2 = a^2 - b^2

to find the distance from the center to each focus.

c = \sqrt{25 - 9} = \sqrt{16} = 4

Answer: The semi-major axis is 5, and each focus is 4 units from the center along the

x

-axis.

Why It Matters

The semi-major axis determines the size and shape of planetary orbits — Kepler's laws describe every planet's path as an ellipse with the Sun at one focus, and

a

directly sets the orbital period. In precalculus courses, identifying the semi-major axis is essential for graphing ellipses and locating their foci.

Common Mistakes

Mistake: Assuming the semi-major axis is always along the

x

-axis.

Correction: The semi-major axis aligns with whichever variable has the larger denominator. If

b^2 > a^2

under

y^2

, the major axis is vertical.