Mathwords logoMathwords

Semiminor Axis — Definition, Formula & Examples

The semiminor axis is the shortest distance from the center of an ellipse to its edge. It is half the length of the minor axis and is typically labeled bb.

For an ellipse with center at the origin and standard form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 where a>b>0a > b > 0, the semiminor axis is the segment of length bb from the center to the ellipse along the direction perpendicular to the major axis.

Key Formula

b=a2−c2b = \sqrt{a^2 - c^2}
Where:
  • bb = Length of the semiminor axis
  • aa = Length of the semimajor axis
  • cc = Distance from the center to each focus

How It Works

In the standard equation of an ellipse, the smaller denominator determines the semiminor axis. If a2a^2 is under x2x^2 and a>ba > b, the major axis runs horizontally and the semiminor axis runs vertically. The semiminor axis, the semimajor axis, and the distance from the center to a focus are connected by the relationship c2=a2−b2c^2 = a^2 - b^2, where cc is the focal distance. A larger bb relative to aa makes the ellipse more circular, while a smaller bb makes it more elongated.

Worked Example

Problem: An ellipse has the equation x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1. Find the length of the semiminor axis.
Identify a² and b²: Since 25 > 9, we have a2=25a^2 = 25 and b2=9b^2 = 9. The semiminor axis corresponds to the smaller value.
a2=25,b2=9a^2 = 25, \quad b^2 = 9
Solve for b: Take the square root of b2b^2 to find the semiminor axis length.
b=9=3b = \sqrt{9} = 3
Answer: The semiminor axis has length b=3b = 3. The ellipse extends 3 units above and below its center along the yy-axis.

Why It Matters

Orbital mechanics relies on the semiminor axis to describe planetary orbits — every planet follows an elliptical path around the sun. In precalculus, you need the semiminor axis to graph ellipses accurately and to calculate eccentricity.

Common Mistakes

Mistake: Assuming the semiminor axis is always along the yy-axis.
Correction: The semiminor axis aligns with whichever variable has the smaller denominator. If x24+y216=1\frac{x^2}{4} + \frac{y^2}{16} = 1, the semiminor axis is along the xx-axis because 4<164 < 16.