Minor Axis of an Ellipse — Definition, Formula & Examples

Q: How do you tell which axis is the minor axis from the equation of an ellipse?

In the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the minor axis corresponds to the smaller denominator. If the smaller denominator is under $y^2$, the minor axis is vertical. If the smaller denominator is under $x^2$, the minor axis is horizontal.

Q: Can the minor axis and major axis be the same length?

Only if the shape is a circle. When $a = b$, the ellipse equation becomes $x^2 + y^2 = a^2$, which is a circle. In a true ellipse, the major axis is always strictly longer than the minor axis.

Minor Axis of an Ellipse

A line through the center of an ellipse which is perpendicular to the major axis. The minor axis is an axis of symmetry.

Ellipse with vertical minor axis labeled, shown as a double-headed arrow through the center, perpendicular to the horizontal...

Key Formula

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{where } a > b

\text{Length of minor axis} = 2b

Where:

$a$ = Semi-major axis — the distance from the center to the farthest point on the ellipse
$b$ = Semi-minor axis — the distance from the center to the nearest point on the ellipse along the minor axis
$2b$ = The full length of the minor axis

Worked Example

Problem: Find the length of the minor axis of the ellipse given by the equation

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

Step 1: Identify

a^2

and

b^2

from the standard form. Since

25 > 9

, we have

a^2 = 25

and

b^2 = 9

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

Step 2: Solve for

b

by taking the square root of

b^2

b = \sqrt{9} = 3

Step 3: Calculate the full length of the minor axis using the formula

2b

\text{Minor axis length} = 2b = 2(3) = 6

Step 4: Identify the endpoints. Because

b^2

is under

y^2

, the minor axis is vertical, running from

(0, -3)

(0, 3)

\text{Endpoints: } (0, -3) \text{ and } (0, 3)

Answer: The minor axis has a length of 6 and runs vertically from

(0, -3)

(0, 3)

Another Example

This example differs because the major axis is vertical (along the $y$ -axis), so the minor axis runs horizontally along the $x$ -axis. It shows students how axis orientation affects which denominator corresponds to $b^2$ .

Problem: An ellipse has a semi-major axis of length 10 along the

y

-axis and a semi-minor axis of length 4 along the

x

-axis. Write the equation of the ellipse and state the length and direction of the minor axis.

Step 1: Since the major axis is along the

y

-axis, the standard form places the larger denominator under

y^2

. Here

a = 10

and

b = 4

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

Step 2: Substitute the values of

a

and

b

\frac{x^2}{16} + \frac{y^2}{100} = 1

Step 3: The minor axis lies along the

x

-axis (perpendicular to the major axis). Its endpoints are

(-4, 0)

and

(4, 0)

\text{Minor axis length} = 2b = 2(4) = 8

Answer: The equation is

\frac{x^2}{16} + \frac{y^2}{100} = 1

. The minor axis is horizontal with length 8, running from

(-4, 0)

(4, 0)

Frequently Asked Questions

What is the difference between the minor axis and the semi-minor axis?

The semi-minor axis is the distance from the center of the ellipse to one end of the minor axis, equal to

b

. The minor axis is the full segment spanning both ends, with length

2b

. Think of the semi-minor axis as half the minor axis.

How do you tell which axis is the minor axis from the equation of an ellipse?

In the standard form

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

, the minor axis corresponds to the smaller denominator. If the smaller denominator is under

y^2

, the minor axis is vertical. If the smaller denominator is under

x^2

, the minor axis is horizontal.

Can the minor axis and major axis be the same length?

Only if the shape is a circle. When

a = b

, the ellipse equation becomes

x^2 + y^2 = a^2

, which is a circle. In a true ellipse, the major axis is always strictly longer than the minor axis.

Minor Axis vs. Major Axis

	Minor Axis	Major Axis
Definition	Shortest diameter of the ellipse, through the center	Longest diameter of the ellipse, through the center
Length formula	$2b$ (where $b < a$ )	$2a$ (where $a > b$ )
Relationship to foci	Perpendicular to the line connecting the foci	Passes through both foci
Symmetry	Acts as an axis of symmetry for the ellipse	Acts as an axis of symmetry for the ellipse
In the standard equation	Corresponds to the smaller denominator	Corresponds to the larger denominator

Why It Matters

The minor axis appears frequently in conic sections courses and standardized tests whenever you need to sketch, classify, or write the equation of an ellipse. In science, it matters for planetary orbits—Kepler's laws describe planets moving along ellipses, and the minor axis determines how "circular" an orbit is. Understanding both axes also helps in engineering and architecture, where elliptical shapes are used in arches, reflectors, and racetracks.

Common Mistakes

Mistake: Confusing

b

with

2b

: writing the semi-minor axis length as the full minor axis length.

Correction: Remember that

b

is only the distance from the center to one endpoint. The full minor axis spans both sides of the center, so its length is

2b

Mistake: Assuming the minor axis is always along the

y

-axis.

Correction: The minor axis is perpendicular to the major axis. If the larger denominator in the standard equation is under

y^2

, the major axis is vertical and the minor axis is horizontal along the

x

-axis. Always compare the denominators to determine orientation.

Related Terms

Major Axis of an Ellipse — The longer axis, perpendicular to the minor axis
Ellipse — The conic section defined by both axes
Axis of Symmetry — The minor axis serves as one axis of symmetry
Perpendicular — Minor and major axes meet at right angles
Minor Diameter of an Ellipse — Another name for the minor axis
Major Diameter of an Ellipse — Another name for the major axis
Line — The minor axis lies along a line through the center