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Semi-Minor Axis — Definition, Formula & Examples

The semi-minor axis is the shorter half-axis of an ellipse, measured from the center to the closest point on the ellipse. It is typically denoted by the variable bb.

For an ellipse with center (h,k)(h, k), the semi-minor axis is the distance bb from the center to either endpoint of the minor axis, where the minor axis is the segment perpendicular to the major axis through the center. By definition, 0<b<a0 < b < a, where aa is the semi-major axis.

Key Formula

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

Worked Example

Problem: Find the semi-minor axis of the ellipse given by x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.
Identify a² and b²: The standard form of an ellipse centered at the origin is x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 where a>ba > b. Here a2=25a^2 = 25 and b2=9b^2 = 9.
a2=25,b2=9a^2 = 25, \quad b^2 = 9
Solve for b: Take the square root of b2b^2 to find the semi-minor axis length.
b=9=3b = \sqrt{9} = 3
Answer: The semi-minor axis is b=3b = 3, so the ellipse extends 3 units above and below the center along the yy-axis.

Why It Matters

You need the semi-minor axis to write the standard equation of an ellipse and to sketch its shape accurately. It also appears in physics when describing planetary orbits, where Kepler's laws use both the semi-major and semi-minor axes to characterize elliptical paths.

Common Mistakes

Mistake: Assuming the denominator under y2y^2 is always b2b^2.
Correction: The larger denominator corresponds to a2a^2 regardless of which variable it sits under. If x29+y225=1\frac{x^2}{9} + \frac{y^2}{25} = 1, then a2=25a^2 = 25 (under y2y^2) and b2=9b^2 = 9 (under x2x^2), making the major axis vertical.