Minor Diameter of an Ellipse

Q: How do you find the minor diameter from the equation of an ellipse?

Write the equation in standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a \geq b$. The smaller denominator is $b^2$. Take the square root to get $b$, then the minor diameter is $2b$. If the larger denominator is under $y^2$ instead of $x^2$, the roles of the axes swap, but you still use the smaller denominator for the minor diameter.

Q: Is the minor diameter of a circle the same as its diameter?

Yes. A circle is a special ellipse where $a = b = r$. Every diameter of a circle has the same length $2r$, so the minor diameter equals the major diameter, and both equal the ordinary diameter of the circle.

Minor Diameter of an Ellipse

The segment through the center of an ellipse perpendicular to the major diameter, or the length of that segment. The minor diameter is the shortest diameter of an ellipse.

Ellipse with a vertical line through its center labeled "minor diameter," representing the shortest diameter perpendicular to...

Key Formula

\text{Minor Diameter} = 2b

Where:

$b$ = The semi-minor axis — the distance from the center of the ellipse to the closest point on the ellipse (the endpoint of the minor axis)
$2b$ = The full minor diameter, measuring the total length of the shortest diameter of the ellipse

Worked Example

Problem: An ellipse has the equation

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

. Find the minor diameter.

Step 1: Identify the standard form of the ellipse equation. The standard form is

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

, where

a > b

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

Step 2: Determine

a^2

and

b^2

. Here

a^2 = 25

and

b^2 = 9

. Since

25 > 9

, the major axis is along the

x

-axis and the minor axis is along the

y

-axis.

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

Step 3: Find

b

by taking the square root of

b^2

b = \sqrt{9} = 3

Step 4: Calculate the minor diameter using the formula

2b

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

Answer: The minor diameter of the ellipse is 6 units.

Another Example

This example derives the minor diameter from the major diameter and focal distance using the relationship $c^2 = a^2 - b^2$ , rather than reading $b$ directly from the equation.

Problem: An ellipse has a major diameter of 20 and the distance from the center to each focus is

c = 8

. Find the minor diameter.

Step 1: Find the semi-major axis

a

from the major diameter.

a = \frac{\text{Major Diameter}}{2} = \frac{20}{2} = 10

Step 2: Use the relationship

c^2 = a^2 - b^2

to solve for

b^2

b^2 = a^2 - c^2 = 10^2 - 8^2 = 100 - 64 = 36

Step 3: Find

b

by taking the square root.

b = \sqrt{36} = 6

Step 4: Compute the minor diameter.

\text{Minor Diameter} = 2b = 2(6) = 12

Answer: The minor diameter is 12 units.

Frequently Asked Questions

What is the difference between the minor diameter and the minor axis of an ellipse?

The terms are often used interchangeably, but there is a subtle distinction. The minor axis refers to the line of symmetry itself (extending infinitely in some definitions) or the full chord through the center, while the minor diameter specifically emphasizes the segment or its measured length,

2b

. In practice, both give the same numerical value.

How do you find the minor diameter from the equation of an ellipse?

Write the equation in standard form

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

where

a \geq b

. The smaller denominator is

b^2

. Take the square root to get

b

, then the minor diameter is

2b

. If the larger denominator is under

y^2

instead of

x^2

, the roles of the axes swap, but you still use the smaller denominator for the minor diameter.

Is the minor diameter of a circle the same as its diameter?

Yes. A circle is a special ellipse where

a = b = r

. Every diameter of a circle has the same length

2r

, so the minor diameter equals the major diameter, and both equal the ordinary diameter of the circle.

Minor Diameter vs. Major Diameter

	Minor Diameter	Major Diameter
Definition	Shortest diameter through the center of the ellipse	Longest diameter through the center of the ellipse
Formula	$2b$	$2a$
Direction	Perpendicular to the major diameter	Passes through both foci
Relationship to foci	Does not pass through the foci	Passes through both foci
When $a = b$ (circle)	Equals the circle's diameter	Equals the circle's diameter

Why It Matters

The minor diameter appears frequently in geometry and precalculus when you work with conic sections. You need it to compute the area of an ellipse (

A = \pi a b

), where

b

is half the minor diameter. It also shows up in engineering and astronomy — for instance, planetary orbits are ellipses, and the minor diameter helps describe how "stretched" or eccentric an orbit is.

Common Mistakes

Mistake: Confusing the minor diameter (

2b

) with the semi-minor axis (

b

Correction: The semi-minor axis is the distance from the center to the ellipse along the shorter direction. The minor diameter is the full distance across, which is twice that value:

2b

Mistake: Assigning the wrong denominator to

b^2

in the standard equation.

Correction: In

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

, the convention is

a \geq b

. Always compare the two denominators: the smaller one is

b^2

(for the minor diameter), regardless of whether it sits under

x^2

y^2

Related Terms

Major Diameter of an Ellipse — The longest diameter, equal to $2a$
Minor Axis of an Ellipse — The axis along the minor diameter
Major Axis of an Ellipse — The axis along the major diameter
Ellipse — The conic section defined by this diameter
Diameter of a Circle or Sphere — Special case when the ellipse is a circle
Perpendicular — Minor diameter is perpendicular to major diameter
Line Segment — The minor diameter is a line segment