Major Diameter of an Ellipse — Formula & Examples

Q: How do you tell which diameter is the major diameter from an equation?

In the standard form $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, the major diameter lies along whichever variable has the larger denominator. If $a^2 > b^2$, the major diameter is horizontal; if $b^2 > a^2$, it is vertical. The major diameter always has length $2 \times \sqrt{\text{larger denominator}}$.

Major Diameter of an Ellipse

The segment joining the vertices of an ellipse, or the length of that segment. The major diameter passes through the center and foci. It is an ellipse's longest diameter.

Ellipse with a horizontal line labeled "major diameter" spanning its full width through the center.

Key Formula

\text{Major Diameter} = 2a

Where:

$a$ = The semi-major axis — the distance from the center of the ellipse to either vertex along the major diameter.

Worked Example

Problem: An ellipse has the equation

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

. Find the major diameter.

Step 1: Recall the standard form of an ellipse centered at the origin:

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

, where

a > b

. The larger denominator sits under the variable along the major diameter.

Step 2: Identify

a^2

and

b^2

. Here

a^2 = 25

and

b^2 = 9

because

25 > 9

a^2 = 25 \implies a = 5

Step 3: Apply the major diameter formula.

\text{Major Diameter} = 2a = 2(5) = 10

Step 4: The vertices (endpoints of the major diameter) are at

(-5, 0)

and

(5, 0)

, confirming the segment has length 10.

Answer: The major diameter is 10 units.

Another Example

This example differs because the ellipse is shifted (not centered at the origin) and the major diameter is vertical rather than horizontal, showing that the larger denominator determines the direction of the major diameter.

Problem: An ellipse has the equation

\dfrac{(x-3)^2}{16} + \dfrac{(y+2)^2}{49} = 1

. Find the major diameter and identify its vertices.

Step 1: Compare the denominators: 16 and 49. Since

49 > 16

, the larger denominator is under the

y

-term, so the major diameter runs vertically.

a^2 = 49 \implies a = 7

Step 2: Compute the major diameter.

\text{Major Diameter} = 2a = 2(7) = 14

Step 3: The center of this ellipse is

(3, -2)

. Because the major diameter is vertical, the vertices are

a = 7

units above and below the center.

\text{Vertices: } (3,\,-2+7) = (3, 5) \quad \text{and} \quad (3,\,-2-7) = (3, -9)

Step 4: Verify: the distance between

(3, 5)

and

(3, -9)

5 - (-9) = 14

. This matches

2a = 14

Answer: The major diameter is 14 units, running vertically from

(3, 5)

(3, -9)

Frequently Asked Questions

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

These terms are often used interchangeably. Strictly, the major axis is the line of symmetry that extends infinitely, while the major diameter is the finite segment (or its length) between the two vertices. In most textbooks and problems, however, 'major axis' refers to the same segment and has the same length

2a

How do you tell which diameter is the major diameter from an equation?

In the standard form

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

, the major diameter lies along whichever variable has the larger denominator. If

a^2 > b^2

, the major diameter is horizontal; if

b^2 > a^2

, it is vertical. The major diameter always has length

2 \times \sqrt{\text{larger denominator}}

Can the major diameter and minor diameter ever be equal?

Only when the ellipse is actually a circle. A circle is a special case where

a = b

, so both diameters have the same length. For a true ellipse (where

a \neq b

), the major diameter is always strictly longer than the minor diameter.

Major Diameter vs. Minor Diameter

	Major Diameter	Minor Diameter
Definition	The longest diameter of the ellipse, connecting the two vertices	The shortest diameter of the ellipse, connecting the two co-vertices
Formula	$2a$ (where $a$ is the semi-major axis)	$2b$ (where $b$ is the semi-minor axis)
Passes through	Center and both foci	Center only (perpendicular to the foci)
Relationship to foci	Foci lie on this diameter, between the center and each vertex	Foci do not lie on this diameter
Standard form clue	Associated with the larger denominator	Associated with the smaller denominator

Why It Matters

The major diameter appears frequently in conic sections courses and standardized tests whenever you work with ellipses. It connects directly to finding the foci (since

c^2 = a^2 - b^2

) and is essential in applications like planetary orbits, where the major diameter determines the longest span of an orbit. Understanding it also helps you sketch ellipses accurately by marking the correct vertices first.

Common Mistakes

Mistake: Confusing

a

with

2a

— reporting the semi-major axis as the full major diameter.

Correction: The value

a

is only half the major diameter. Always multiply by 2: Major Diameter

= 2a

. If

a^2 = 25

, the major diameter is

2(5) = 10

, not 5.

Mistake: Assuming the major diameter is always horizontal (along the

x

-axis).

Correction: The major diameter aligns with whichever variable has the larger denominator in the standard equation. If the larger denominator is under

y^2

, the major diameter is vertical.

Related Terms

Ellipse — The conic section this diameter belongs to
Minor Diameter of an Ellipse — The shorter diameter, perpendicular to the major
Major Axis of an Ellipse — Often used interchangeably with major diameter
Vertices of an Ellipse — Endpoints of the major diameter
Foci of an Ellipse — Two interior points lying on the major diameter
Diameter of a Circle or Sphere — Analogous concept for circles where all diameters are equal
Line Segment — The geometric object that the major diameter is