Semimajor Axis — Definition, Formula & Examples

The semimajor axis is half the length of the major axis 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 semimajor axis is the value $a$ , equal to the distance from the center to either vertex along the major axis. In the standard form equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$ , the semimajor axis is $a$ .

Key Formula

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

Where:

$a$ = Semimajor axis (distance from center to vertex along the major axis)

How It Works

In the standard equation of an ellipse, the semimajor axis

a

always corresponds to the larger denominator (after taking the square root). If the larger denominator is under

x^2

, the major axis is horizontal; if it is under

y^2

, the major axis is vertical. The semimajor axis also determines the eccentricity and the location of 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 semimajor axis of the ellipse given by the equation

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

Identify the larger denominator: The denominators are 25 and 9. Since

25 > 9

, the semimajor axis is associated with the

x^2

term.

a^2 = 25

Take the square root: Solve for

a

by taking the positive square root.

a = \sqrt{25} = 5

Answer: The semimajor axis is

a = 5

, and the major axis is horizontal.

Why It Matters

In astronomy, the semimajor axis of a planet's orbit determines its average distance from the Sun and is directly linked to its orbital period through Kepler's third law. In precalculus and analytic geometry courses, knowing how to identify the semimajor axis is essential for graphing ellipses and solving conic section problems.

Common Mistakes

Mistake: Confusing the semimajor axis with the full major axis length.

Correction: The semimajor axis is half the major axis. If the major axis has length 10, the semimajor axis

a

is 5, not 10.